Step |
Hyp |
Ref |
Expression |
1 |
|
ptcmp.1 |
⊢ 𝑆 = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
2 |
|
ptcmp.2 |
⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) |
3 |
|
ptcmp.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
ptcmp.4 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Comp ) |
5 |
|
ptcmp.5 |
⊢ ( 𝜑 → 𝑋 ∈ ( UFL ∩ dom card ) ) |
6 |
|
ptcmplem2.5 |
⊢ ( 𝜑 → 𝑈 ⊆ ran 𝑆 ) |
7 |
|
ptcmplem2.6 |
⊢ ( 𝜑 → 𝑋 = ∪ 𝑈 ) |
8 |
|
ptcmplem2.7 |
⊢ ( 𝜑 → ¬ ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) |
9 |
|
ptcmplem3.8 |
⊢ 𝐾 = { 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ∣ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝑈 } |
10 |
|
rabexg |
⊢ ( 𝐴 ∈ 𝑉 → { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∈ V ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∈ V ) |
12 |
1 2 3 4 5 6 7 8
|
ptcmplem2 |
⊢ ( 𝜑 → ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∪ ( 𝐹 ‘ 𝑘 ) ∈ dom card ) |
13 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) → 𝑦 ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) |
14 |
13
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ V ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → 𝑦 ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) |
15 |
14
|
rabssdv |
⊢ ( 𝜑 → { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
16 |
15
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
17 |
|
ss2iun |
⊢ ( ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ ( 𝐹 ‘ 𝑘 ) → ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∪ ( 𝐹 ‘ 𝑘 ) ) |
18 |
16 17
|
syl |
⊢ ( 𝜑 → ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∪ ( 𝐹 ‘ 𝑘 ) ) |
19 |
|
ssnum |
⊢ ( ( ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∪ ( 𝐹 ‘ 𝑘 ) ∈ dom card ∧ ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ⊆ ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∪ ( 𝐹 ‘ 𝑘 ) ) → ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ∈ dom card ) |
20 |
12 18 19
|
syl2anc |
⊢ ( 𝜑 → ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ∈ dom card ) |
21 |
|
elrabi |
⊢ ( 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } → 𝑘 ∈ 𝐴 ) |
22 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) |
23 |
|
ssdif0 |
⊢ ( ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ↔ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) = ∅ ) |
24 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑘 ) ∈ Comp ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ( 𝐹 ‘ 𝑘 ) ∈ Comp ) |
26 |
9
|
ssrab3 |
⊢ 𝐾 ⊆ ( 𝐹 ‘ 𝑘 ) |
27 |
26
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → 𝐾 ⊆ ( 𝐹 ‘ 𝑘 ) ) |
28 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) |
29 |
|
uniss |
⊢ ( 𝐾 ⊆ ( 𝐹 ‘ 𝑘 ) → ∪ 𝐾 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
30 |
26 29
|
mp1i |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ∪ 𝐾 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
31 |
28 30
|
eqssd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐾 ) |
32 |
|
eqid |
⊢ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) |
33 |
32
|
cmpcov |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) ∈ Comp ∧ 𝐾 ⊆ ( 𝐹 ‘ 𝑘 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐾 ) → ∃ 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) |
34 |
25 27 31 33
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ∃ 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) |
35 |
|
elfpw |
⊢ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ↔ ( 𝑡 ⊆ 𝐾 ∧ 𝑡 ∈ Fin ) ) |
36 |
35
|
simplbi |
⊢ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) → 𝑡 ⊆ 𝐾 ) |
37 |
36
|
ad2antrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑡 ⊆ 𝐾 ) |
38 |
37
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑥 ∈ 𝑡 ) → 𝑥 ∈ 𝐾 ) |
39 |
|
imaeq2 |
⊢ ( 𝑢 = 𝑥 → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
40 |
39
|
eleq1d |
⊢ ( 𝑢 = 𝑥 → ( ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝑈 ↔ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 ) ) |
41 |
40 9
|
elrab2 |
⊢ ( 𝑥 ∈ 𝐾 ↔ ( 𝑥 ∈ ( 𝐹 ‘ 𝑘 ) ∧ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 ) ) |
42 |
41
|
simprbi |
⊢ ( 𝑥 ∈ 𝐾 → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 ) |
43 |
38 42
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑥 ∈ 𝑡 ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 ) |
44 |
43
|
fmpttd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) : 𝑡 ⟶ 𝑈 ) |
45 |
44
|
frnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ⊆ 𝑈 ) |
46 |
35
|
simprbi |
⊢ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) → 𝑡 ∈ Fin ) |
47 |
46
|
ad2antrl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑡 ∈ Fin ) |
48 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) = ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
49 |
48
|
rnmpt |
⊢ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) = { 𝑓 ∣ ∃ 𝑥 ∈ 𝑡 𝑓 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) } |
50 |
|
abrexfi |
⊢ ( 𝑡 ∈ Fin → { 𝑓 ∣ ∃ 𝑥 ∈ 𝑡 𝑓 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) } ∈ Fin ) |
51 |
49 50
|
eqeltrid |
⊢ ( 𝑡 ∈ Fin → ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ Fin ) |
52 |
47 51
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ Fin ) |
53 |
|
elfpw |
⊢ ( ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ ( 𝒫 𝑈 ∩ Fin ) ↔ ( ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ⊆ 𝑈 ∧ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ Fin ) ) |
54 |
45 52 53
|
sylanbrc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ ( 𝒫 𝑈 ∩ Fin ) ) |
55 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑓 ‘ 𝑛 ) = ( 𝑓 ‘ 𝑘 ) ) |
56 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑘 ) ) |
57 |
56
|
unieqd |
⊢ ( 𝑛 = 𝑘 → ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑘 ) ) |
58 |
55 57
|
eleq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝑓 ‘ 𝑛 ) ∈ ∪ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝑓 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
59 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ 𝑋 ) |
60 |
59 2
|
eleqtrdi |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ) |
61 |
|
vex |
⊢ 𝑓 ∈ V |
62 |
61
|
elixp |
⊢ ( 𝑓 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ↔ ( 𝑓 Fn 𝐴 ∧ ∀ 𝑛 ∈ 𝐴 ( 𝑓 ‘ 𝑛 ) ∈ ∪ ( 𝐹 ‘ 𝑛 ) ) ) |
63 |
62
|
simprbi |
⊢ ( 𝑓 ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) → ∀ 𝑛 ∈ 𝐴 ( 𝑓 ‘ 𝑛 ) ∈ ∪ ( 𝐹 ‘ 𝑛 ) ) |
64 |
60 63
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ∀ 𝑛 ∈ 𝐴 ( 𝑓 ‘ 𝑛 ) ∈ ∪ ( 𝐹 ‘ 𝑛 ) ) |
65 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → 𝑘 ∈ 𝐴 ) |
66 |
58 64 65
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) |
67 |
|
simplrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) |
68 |
66 67
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ( 𝑓 ‘ 𝑘 ) ∈ ∪ 𝑡 ) |
69 |
|
eluni2 |
⊢ ( ( 𝑓 ‘ 𝑘 ) ∈ ∪ 𝑡 ↔ ∃ 𝑥 ∈ 𝑡 ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) |
70 |
68 69
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ∃ 𝑥 ∈ 𝑡 ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) |
71 |
|
fveq1 |
⊢ ( 𝑤 = 𝑓 → ( 𝑤 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑘 ) ) |
72 |
71
|
eleq1d |
⊢ ( 𝑤 = 𝑓 → ( ( 𝑤 ‘ 𝑘 ) ∈ 𝑥 ↔ ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) ) |
73 |
|
eqid |
⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) |
74 |
73
|
mptpreima |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) = { 𝑤 ∈ 𝑋 ∣ ( 𝑤 ‘ 𝑘 ) ∈ 𝑥 } |
75 |
72 74
|
elrab2 |
⊢ ( 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ↔ ( 𝑓 ∈ 𝑋 ∧ ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) ) |
76 |
75
|
baib |
⊢ ( 𝑓 ∈ 𝑋 → ( 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ↔ ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) ) |
77 |
76
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) ∧ 𝑥 ∈ 𝑡 ) → ( 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ↔ ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) ) |
78 |
77
|
rexbidva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ 𝑡 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ↔ ∃ 𝑥 ∈ 𝑡 ( 𝑓 ‘ 𝑘 ) ∈ 𝑥 ) ) |
79 |
70 78
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → ∃ 𝑥 ∈ 𝑡 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
80 |
|
eliun |
⊢ ( 𝑓 ∈ ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ↔ ∃ 𝑥 ∈ 𝑡 𝑓 ∈ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
81 |
79 80
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) ∧ 𝑓 ∈ 𝑋 ) → 𝑓 ∈ ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
82 |
81
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ( 𝑓 ∈ 𝑋 → 𝑓 ∈ ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) |
83 |
82
|
ssrdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑋 ⊆ ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) |
84 |
43
|
ralrimiva |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∀ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 ) |
85 |
|
dfiun2g |
⊢ ( ∀ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ∈ 𝑈 → ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ 𝑡 𝑓 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) } ) |
86 |
84 85
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ 𝑡 𝑓 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) } ) |
87 |
49
|
unieqi |
⊢ ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ 𝑡 𝑓 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) } |
88 |
86 87
|
eqtr4di |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∪ 𝑥 ∈ 𝑡 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) = ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) |
89 |
83 88
|
sseqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑋 ⊆ ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) |
90 |
45
|
unissd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ⊆ ∪ 𝑈 ) |
91 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑋 = ∪ 𝑈 ) |
92 |
90 91
|
sseqtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ⊆ 𝑋 ) |
93 |
89 92
|
eqssd |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → 𝑋 = ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) |
94 |
|
unieq |
⊢ ( 𝑧 = ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) → ∪ 𝑧 = ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) |
95 |
94
|
rspceeqv |
⊢ ( ( ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ∈ ( 𝒫 𝑈 ∩ Fin ) ∧ 𝑋 = ∪ ran ( 𝑥 ∈ 𝑡 ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑥 ) ) ) → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) |
96 |
54 93 95
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) ∧ ( 𝑡 ∈ ( 𝒫 𝐾 ∩ Fin ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝑡 ) ) → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) |
97 |
34 96
|
rexlimddv |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 ) → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) |
98 |
97
|
ex |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ 𝐾 → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) ) |
99 |
23 98
|
syl5bir |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) = ∅ → ∃ 𝑧 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑋 = ∪ 𝑧 ) ) |
100 |
22 99
|
mtod |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) = ∅ ) |
101 |
|
neq0 |
⊢ ( ¬ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) = ∅ ↔ ∃ 𝑦 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
102 |
100 101
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ∃ 𝑦 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
103 |
|
rexv |
⊢ ( ∃ 𝑦 ∈ V 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ ∃ 𝑦 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
104 |
102 103
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ∃ 𝑦 ∈ V 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
105 |
21 104
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ) → ∃ 𝑦 ∈ V 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
106 |
105
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∃ 𝑦 ∈ V 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
107 |
|
eleq1 |
⊢ ( 𝑦 = ( 𝑔 ‘ 𝑘 ) → ( 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
108 |
107
|
ac6num |
⊢ ( ( { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∈ V ∧ ∪ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } { 𝑦 ∈ V ∣ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) } ∈ dom card ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ∃ 𝑦 ∈ V 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∃ 𝑔 ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
109 |
11 20 106 108
|
syl3anc |
⊢ ( 𝜑 → ∃ 𝑔 ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
110 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) → 𝐴 ∈ 𝑉 ) |
111 |
110
|
mptexd |
⊢ ( ( 𝜑 ∧ ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) → ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ∈ V ) |
112 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑚 ) ∈ V |
113 |
112
|
uniex |
⊢ ∪ ( 𝐹 ‘ 𝑚 ) ∈ V |
114 |
113
|
uniex |
⊢ ∪ ∪ ( 𝐹 ‘ 𝑚 ) ∈ V |
115 |
|
fvex |
⊢ ( 𝑔 ‘ 𝑚 ) ∈ V |
116 |
114 115
|
ifex |
⊢ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ∈ V |
117 |
116
|
rgenw |
⊢ ∀ 𝑚 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ∈ V |
118 |
|
eqid |
⊢ ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) |
119 |
118
|
fnmpt |
⊢ ( ∀ 𝑚 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ∈ V → ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ) |
120 |
117 119
|
mp1i |
⊢ ( ( 𝜑 ∧ ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) → ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ) |
121 |
57
|
breq1d |
⊢ ( 𝑛 = 𝑘 → ( ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o ↔ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) |
122 |
121
|
notbid |
⊢ ( 𝑛 = 𝑘 → ( ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o ↔ ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) |
123 |
122
|
ralrab |
⊢ ( ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ ∀ 𝑘 ∈ 𝐴 ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
124 |
|
iftrue |
⊢ ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) = ∪ ∪ ( 𝐹 ‘ 𝑘 ) ) |
125 |
124
|
ad2antll |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) = ∪ ∪ ( 𝐹 ‘ 𝑘 ) ) |
126 |
102
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → ∃ 𝑦 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
127 |
13
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → 𝑦 ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) |
128 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) |
129 |
|
en1b |
⊢ ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ↔ ∪ ( 𝐹 ‘ 𝑘 ) = { ∪ ∪ ( 𝐹 ‘ 𝑘 ) } ) |
130 |
128 129
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∪ ( 𝐹 ‘ 𝑘 ) = { ∪ ∪ ( 𝐹 ‘ 𝑘 ) } ) |
131 |
127 130
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → 𝑦 ∈ { ∪ ∪ ( 𝐹 ‘ 𝑘 ) } ) |
132 |
|
elsni |
⊢ ( 𝑦 ∈ { ∪ ∪ ( 𝐹 ‘ 𝑘 ) } → 𝑦 = ∪ ∪ ( 𝐹 ‘ 𝑘 ) ) |
133 |
131 132
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → 𝑦 = ∪ ∪ ( 𝐹 ‘ 𝑘 ) ) |
134 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
135 |
133 134
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) ∧ 𝑦 ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∪ ∪ ( 𝐹 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
136 |
126 135
|
exlimddv |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → ∪ ∪ ( 𝐹 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
137 |
136
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → ∪ ∪ ( 𝐹 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
138 |
125 137
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
139 |
138
|
a1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ ( 𝑘 ∈ 𝐴 ∧ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) → ( ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
140 |
139
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ 𝑘 ∈ 𝐴 ) → ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) ) |
141 |
|
pm2.27 |
⊢ ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
142 |
|
iffalse |
⊢ ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) = ( 𝑔 ‘ 𝑘 ) ) |
143 |
142
|
eleq1d |
⊢ ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
144 |
141 143
|
sylibrd |
⊢ ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
145 |
140 144
|
pm2.61d1 |
⊢ ( ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) ∧ 𝑘 ∈ 𝐴 ) → ( ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
146 |
145
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) → ( ∀ 𝑘 ∈ 𝐴 ( ¬ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o → ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
147 |
123 146
|
syl5bi |
⊢ ( ( 𝜑 ∧ 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ) → ( ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) → ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
148 |
147
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) → ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) |
149 |
|
fneq1 |
⊢ ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) → ( 𝑓 Fn 𝐴 ↔ ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ) ) |
150 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) → ( 𝑓 ‘ 𝑘 ) = ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ‘ 𝑘 ) ) |
151 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑘 ) ) |
152 |
151
|
unieqd |
⊢ ( 𝑚 = 𝑘 → ∪ ( 𝐹 ‘ 𝑚 ) = ∪ ( 𝐹 ‘ 𝑘 ) ) |
153 |
152
|
breq1d |
⊢ ( 𝑚 = 𝑘 → ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o ↔ ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o ) ) |
154 |
152
|
unieqd |
⊢ ( 𝑚 = 𝑘 → ∪ ∪ ( 𝐹 ‘ 𝑚 ) = ∪ ∪ ( 𝐹 ‘ 𝑘 ) ) |
155 |
|
fveq2 |
⊢ ( 𝑚 = 𝑘 → ( 𝑔 ‘ 𝑚 ) = ( 𝑔 ‘ 𝑘 ) ) |
156 |
153 154 155
|
ifbieq12d |
⊢ ( 𝑚 = 𝑘 → if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) = if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ) |
157 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑘 ) ∈ V |
158 |
157
|
uniex |
⊢ ∪ ( 𝐹 ‘ 𝑘 ) ∈ V |
159 |
158
|
uniex |
⊢ ∪ ∪ ( 𝐹 ‘ 𝑘 ) ∈ V |
160 |
|
fvex |
⊢ ( 𝑔 ‘ 𝑘 ) ∈ V |
161 |
159 160
|
ifex |
⊢ if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ V |
162 |
156 118 161
|
fvmpt |
⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ‘ 𝑘 ) = if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ) |
163 |
150 162
|
sylan9eq |
⊢ ( ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑘 ) = if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ) |
164 |
163
|
eleq1d |
⊢ ( ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
165 |
164
|
ralbidva |
⊢ ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) → ( ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ↔ ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
166 |
149 165
|
anbi12d |
⊢ ( 𝑓 = ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) → ( ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ↔ ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) ) |
167 |
166
|
spcegv |
⊢ ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ∈ V → ( ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) ) |
168 |
167
|
3impib |
⊢ ( ( ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) ∈ V ∧ ( 𝑚 ∈ 𝐴 ↦ if ( ∪ ( 𝐹 ‘ 𝑚 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑚 ) , ( 𝑔 ‘ 𝑚 ) ) ) Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 if ( ∪ ( 𝐹 ‘ 𝑘 ) ≈ 1o , ∪ ∪ ( 𝐹 ‘ 𝑘 ) , ( 𝑔 ‘ 𝑘 ) ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
169 |
111 120 148 168
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑔 : { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ⟶ V ∧ ∀ 𝑘 ∈ { 𝑛 ∈ 𝐴 ∣ ¬ ∪ ( 𝐹 ‘ 𝑛 ) ≈ 1o } ( 𝑔 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |
170 |
109 169
|
exlimddv |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ ∪ 𝐾 ) ) ) |