Step |
Hyp |
Ref |
Expression |
1 |
|
fnwe2.su |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝑥 ) → 𝑆 = 𝑈 ) |
2 |
|
fnwe2.t |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ∨ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ∧ 𝑥 𝑈 𝑦 ) ) } |
3 |
|
fnwe2.s |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑈 We { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) } ) |
4 |
|
fnwe2.f |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ 𝐵 ) |
5 |
|
fnwe2.r |
⊢ ( 𝜑 → 𝑅 We 𝐵 ) |
6 |
|
fnwe2lem2.a |
⊢ ( 𝜑 → 𝑎 ⊆ 𝐴 ) |
7 |
|
fnwe2lem2.n0 |
⊢ ( 𝜑 → 𝑎 ≠ ∅ ) |
8 |
|
ffun |
⊢ ( ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ 𝐵 → Fun ( 𝐹 ↾ 𝐴 ) ) |
9 |
|
vex |
⊢ 𝑎 ∈ V |
10 |
9
|
funimaex |
⊢ ( Fun ( 𝐹 ↾ 𝐴 ) → ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∈ V ) |
11 |
4 8 10
|
3syl |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∈ V ) |
12 |
|
wefr |
⊢ ( 𝑅 We 𝐵 → 𝑅 Fr 𝐵 ) |
13 |
5 12
|
syl |
⊢ ( 𝜑 → 𝑅 Fr 𝐵 ) |
14 |
|
imassrn |
⊢ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ⊆ ran ( 𝐹 ↾ 𝐴 ) |
15 |
4
|
frnd |
⊢ ( 𝜑 → ran ( 𝐹 ↾ 𝐴 ) ⊆ 𝐵 ) |
16 |
14 15
|
sstrid |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ⊆ 𝐵 ) |
17 |
|
incom |
⊢ ( dom ( 𝐹 ↾ 𝐴 ) ∩ 𝑎 ) = ( 𝑎 ∩ dom ( 𝐹 ↾ 𝐴 ) ) |
18 |
4
|
fdmd |
⊢ ( 𝜑 → dom ( 𝐹 ↾ 𝐴 ) = 𝐴 ) |
19 |
6 18
|
sseqtrrd |
⊢ ( 𝜑 → 𝑎 ⊆ dom ( 𝐹 ↾ 𝐴 ) ) |
20 |
|
df-ss |
⊢ ( 𝑎 ⊆ dom ( 𝐹 ↾ 𝐴 ) ↔ ( 𝑎 ∩ dom ( 𝐹 ↾ 𝐴 ) ) = 𝑎 ) |
21 |
19 20
|
sylib |
⊢ ( 𝜑 → ( 𝑎 ∩ dom ( 𝐹 ↾ 𝐴 ) ) = 𝑎 ) |
22 |
17 21
|
syl5eq |
⊢ ( 𝜑 → ( dom ( 𝐹 ↾ 𝐴 ) ∩ 𝑎 ) = 𝑎 ) |
23 |
22 7
|
eqnetrd |
⊢ ( 𝜑 → ( dom ( 𝐹 ↾ 𝐴 ) ∩ 𝑎 ) ≠ ∅ ) |
24 |
|
imadisj |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) = ∅ ↔ ( dom ( 𝐹 ↾ 𝐴 ) ∩ 𝑎 ) = ∅ ) |
25 |
24
|
necon3bii |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ≠ ∅ ↔ ( dom ( 𝐹 ↾ 𝐴 ) ∩ 𝑎 ) ≠ ∅ ) |
26 |
23 25
|
sylibr |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ≠ ∅ ) |
27 |
|
fri |
⊢ ( ( ( ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∈ V ∧ 𝑅 Fr 𝐵 ) ∧ ( ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ⊆ 𝐵 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ≠ ∅ ) ) → ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ) |
28 |
11 13 16 26 27
|
syl22anc |
⊢ ( 𝜑 → ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ) |
29 |
|
df-ima |
⊢ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) = ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) |
30 |
29
|
rexeqi |
⊢ ( ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∃ 𝑑 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ) |
31 |
4
|
ffnd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ) |
32 |
|
fnssres |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ∧ 𝑎 ⊆ 𝐴 ) → ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) Fn 𝑎 ) |
33 |
31 6 32
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) Fn 𝑎 ) |
34 |
|
breq2 |
⊢ ( 𝑑 = ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) → ( 𝑒 𝑅 𝑑 ↔ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
35 |
34
|
notbid |
⊢ ( 𝑑 = ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) → ( ¬ 𝑒 𝑅 𝑑 ↔ ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
36 |
35
|
ralbidv |
⊢ ( 𝑑 = ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) → ( ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
37 |
36
|
rexrn |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) Fn 𝑎 → ( ∃ 𝑑 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∃ 𝑓 ∈ 𝑎 ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
38 |
33 37
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∃ 𝑓 ∈ 𝑎 ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
39 |
30 38
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∃ 𝑓 ∈ 𝑎 ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
40 |
29
|
raleqi |
⊢ ( ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑒 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) |
41 |
|
breq1 |
⊢ ( 𝑒 = ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) → ( 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
42 |
41
|
notbid |
⊢ ( 𝑒 = ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) → ( ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
43 |
42
|
ralrn |
⊢ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) Fn 𝑎 → ( ∀ 𝑒 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
44 |
33 43
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑒 ∈ ran ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
45 |
40 44
|
syl5bb |
⊢ ( 𝜑 → ( ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) → ( ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ) ) |
47 |
6
|
resabs1d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) = ( 𝐹 ↾ 𝑎 ) ) |
48 |
47
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) = ( 𝐹 ↾ 𝑎 ) ) |
49 |
48
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) = ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑑 ) ) |
50 |
|
fvres |
⊢ ( 𝑑 ∈ 𝑎 → ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑑 ) = ( 𝐹 ‘ 𝑑 ) ) |
51 |
50
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑑 ) = ( 𝐹 ‘ 𝑑 ) ) |
52 |
49 51
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) = ( 𝐹 ‘ 𝑑 ) ) |
53 |
48
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) = ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑓 ) ) |
54 |
|
fvres |
⊢ ( 𝑓 ∈ 𝑎 → ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑓 ) = ( 𝐹 ‘ 𝑓 ) ) |
55 |
54
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( 𝐹 ↾ 𝑎 ) ‘ 𝑓 ) = ( 𝐹 ‘ 𝑓 ) ) |
56 |
53 55
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) = ( 𝐹 ‘ 𝑓 ) ) |
57 |
52 56
|
breq12d |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
58 |
57
|
notbid |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) ∧ 𝑑 ∈ 𝑎 ) → ( ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
59 |
58
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) → ( ∀ 𝑑 ∈ 𝑎 ¬ ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑑 ) 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
60 |
46 59
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) → ( ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
61 |
60
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝑎 ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 ( ( ( 𝐹 ↾ 𝐴 ) ↾ 𝑎 ) ‘ 𝑓 ) ↔ ∃ 𝑓 ∈ 𝑎 ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
62 |
39 61
|
bitrd |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 ↔ ∃ 𝑓 ∈ 𝑎 ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
63 |
9
|
inex1 |
⊢ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∈ V |
64 |
63
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∈ V ) |
65 |
6
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) → 𝑓 ∈ 𝐴 ) |
66 |
1 2 3
|
fnwe2lem1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 We { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
67 |
|
wefr |
⊢ ( ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 We { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 Fr { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
68 |
66 67
|
syl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 Fr { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
69 |
65 68
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑎 ) → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 Fr { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
70 |
69
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 Fr { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
71 |
|
inss2 |
⊢ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ⊆ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } |
72 |
71
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ⊆ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
73 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → 𝑓 ∈ 𝑎 ) |
74 |
|
fveqeq2 |
⊢ ( 𝑦 = 𝑓 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑓 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
75 |
65
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → 𝑓 ∈ 𝐴 ) |
76 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( 𝐹 ‘ 𝑓 ) = ( 𝐹 ‘ 𝑓 ) ) |
77 |
74 75 76
|
elrabd |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → 𝑓 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) |
78 |
73 77
|
elind |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → 𝑓 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ) |
79 |
78
|
ne0d |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ≠ ∅ ) |
80 |
|
fri |
⊢ ( ( ( ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∈ V ∧ ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 Fr { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∧ ( ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ⊆ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ∧ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ≠ ∅ ) ) → ∃ 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) |
81 |
64 70 72 79 80
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ∃ 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) |
82 |
|
elin |
⊢ ( 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑒 ∈ 𝑎 ∧ 𝑒 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ) |
83 |
|
fveqeq2 |
⊢ ( 𝑦 = 𝑒 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
84 |
83
|
elrab |
⊢ ( 𝑒 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ↔ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
85 |
84
|
anbi2i |
⊢ ( ( 𝑒 ∈ 𝑎 ∧ 𝑒 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) |
86 |
82 85
|
bitri |
⊢ ( 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) |
87 |
|
elin |
⊢ ( 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑔 ∈ 𝑎 ∧ 𝑔 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ) |
88 |
|
fveqeq2 |
⊢ ( 𝑦 = 𝑔 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
89 |
88
|
elrab |
⊢ ( 𝑔 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ↔ ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
90 |
89
|
anbi2i |
⊢ ( ( 𝑔 ∈ 𝑎 ∧ 𝑔 ∈ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑔 ∈ 𝑎 ∧ ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) |
91 |
87 90
|
bitri |
⊢ ( 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ↔ ( 𝑔 ∈ 𝑎 ∧ ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) |
92 |
91
|
imbi1i |
⊢ ( ( 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ↔ ( ( 𝑔 ∈ 𝑎 ∧ ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
93 |
|
impexp |
⊢ ( ( ( 𝑔 ∈ 𝑎 ∧ ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ↔ ( 𝑔 ∈ 𝑎 → ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
94 |
92 93
|
bitri |
⊢ ( ( 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ↔ ( 𝑔 ∈ 𝑎 → ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
95 |
94
|
ralbii2 |
⊢ ( ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ↔ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
96 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) → 𝑒 ∈ 𝑎 ) |
97 |
|
fveq2 |
⊢ ( 𝑑 = 𝑐 → ( 𝐹 ‘ 𝑑 ) = ( 𝐹 ‘ 𝑐 ) ) |
98 |
97
|
breq1d |
⊢ ( 𝑑 = 𝑐 → ( ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
99 |
98
|
notbid |
⊢ ( 𝑑 = 𝑐 → ( ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ↔ ¬ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
100 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) → ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) |
101 |
100
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) |
102 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → 𝑐 ∈ 𝑎 ) |
103 |
99 101 102
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ¬ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) |
104 |
|
simprrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) → ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) |
105 |
104
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) |
106 |
105
|
breq2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ( ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ↔ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) |
107 |
103 106
|
mtbird |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ¬ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ) |
108 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) → 𝑎 ⊆ 𝐴 ) |
109 |
108
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → 𝑐 ∈ 𝐴 ) |
110 |
109
|
adantrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → 𝑐 ∈ 𝐴 ) |
111 |
|
simprr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) |
112 |
104
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) |
113 |
111 112
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑓 ) ) |
114 |
|
eleq1w |
⊢ ( 𝑔 = 𝑐 → ( 𝑔 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴 ) ) |
115 |
|
fveqeq2 |
⊢ ( 𝑔 = 𝑐 → ( ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ↔ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑓 ) ) ) |
116 |
114 115
|
anbi12d |
⊢ ( 𝑔 = 𝑐 → ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) ↔ ( 𝑐 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) |
117 |
|
breq1 |
⊢ ( 𝑔 = 𝑐 → ( 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ↔ 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
118 |
117
|
notbid |
⊢ ( 𝑔 = 𝑐 → ( ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ↔ ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
119 |
116 118
|
imbi12d |
⊢ ( 𝑔 = 𝑐 → ( ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ↔ ( ( 𝑐 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
120 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
121 |
|
simprl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → 𝑐 ∈ 𝑎 ) |
122 |
119 120 121
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( ( 𝑐 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
123 |
110 113 122
|
mp2and |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) |
124 |
111 112
|
eqtr2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( 𝐹 ‘ 𝑓 ) = ( 𝐹 ‘ 𝑐 ) ) |
125 |
124
|
csbeq1d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 ) |
126 |
125
|
breqd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ( 𝑐 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ↔ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
127 |
123 126
|
mtbid |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ ( 𝑐 ∈ 𝑎 ∧ ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ) ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) |
128 |
127
|
expr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
129 |
|
imnan |
⊢ ( ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) → ¬ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ↔ ¬ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
130 |
128 129
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ¬ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) |
131 |
|
ioran |
⊢ ( ¬ ( ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ∨ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ↔ ( ¬ ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ∧ ¬ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
132 |
107 130 131
|
sylanbrc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ¬ ( ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ∨ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
133 |
1 2
|
fnwe2val |
⊢ ( 𝑐 𝑇 𝑒 ↔ ( ( 𝐹 ‘ 𝑐 ) 𝑅 ( 𝐹 ‘ 𝑒 ) ∨ ( ( 𝐹 ‘ 𝑐 ) = ( 𝐹 ‘ 𝑒 ) ∧ 𝑐 ⦋ ( 𝐹 ‘ 𝑐 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ) |
134 |
132 133
|
sylnibr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) ∧ 𝑐 ∈ 𝑎 ) → ¬ 𝑐 𝑇 𝑒 ) |
135 |
134
|
ralrimiva |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) → ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑒 ) |
136 |
|
breq2 |
⊢ ( 𝑏 = 𝑒 → ( 𝑐 𝑇 𝑏 ↔ 𝑐 𝑇 𝑒 ) ) |
137 |
136
|
notbid |
⊢ ( 𝑏 = 𝑒 → ( ¬ 𝑐 𝑇 𝑏 ↔ ¬ 𝑐 𝑇 𝑒 ) ) |
138 |
137
|
ralbidv |
⊢ ( 𝑏 = 𝑒 → ( ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ↔ ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑒 ) ) |
139 |
138
|
rspcev |
⊢ ( ( 𝑒 ∈ 𝑎 ∧ ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑒 ) → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) |
140 |
96 135 139
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) ∧ ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) ) → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) |
141 |
140
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) → ( ∀ 𝑔 ∈ 𝑎 ( ( 𝑔 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑔 ) = ( 𝐹 ‘ 𝑓 ) ) → ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 ) → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) |
142 |
95 141
|
syl5bi |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) ∧ ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) ) → ( ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) |
143 |
142
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( ( 𝑒 ∈ 𝑎 ∧ ( 𝑒 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑒 ) = ( 𝐹 ‘ 𝑓 ) ) ) → ( ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) ) |
144 |
86 143
|
syl5bi |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) → ( ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) ) |
145 |
144
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ( ∃ 𝑒 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ∀ 𝑔 ∈ ( 𝑎 ∩ { 𝑦 ∈ 𝐴 ∣ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑓 ) } ) ¬ 𝑔 ⦋ ( 𝐹 ‘ 𝑓 ) / 𝑧 ⦌ 𝑆 𝑒 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) |
146 |
81 145
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑎 ∧ ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) ) ) → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) |
147 |
146
|
rexlimdvaa |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝑎 ∀ 𝑑 ∈ 𝑎 ¬ ( 𝐹 ‘ 𝑑 ) 𝑅 ( 𝐹 ‘ 𝑓 ) → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) |
148 |
62 147
|
sylbid |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ∀ 𝑒 ∈ ( ( 𝐹 ↾ 𝐴 ) “ 𝑎 ) ¬ 𝑒 𝑅 𝑑 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) ) |
149 |
28 148
|
mpd |
⊢ ( 𝜑 → ∃ 𝑏 ∈ 𝑎 ∀ 𝑐 ∈ 𝑎 ¬ 𝑐 𝑇 𝑏 ) |