Step |
Hyp |
Ref |
Expression |
1 |
|
acunirnmpt.0 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
acunirnmpt.1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ≠ ∅ ) |
3 |
|
acunirnmpt.2 |
⊢ 𝐶 = ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) |
4 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
5 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝜑 ) |
6 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑗 ∈ 𝐴 ) |
7 |
5 6 2
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝐵 ≠ ∅ ) |
8 |
4 7
|
eqnetrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑦 ≠ ∅ ) |
9 |
3
|
eleq2i |
⊢ ( 𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) |
10 |
|
vex |
⊢ 𝑦 ∈ V |
11 |
|
eqid |
⊢ ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) |
12 |
11
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) ) |
13 |
10 12
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) |
14 |
9 13
|
bitri |
⊢ ( 𝑦 ∈ 𝐶 ↔ ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) |
15 |
14
|
biimpi |
⊢ ( 𝑦 ∈ 𝐶 → ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) |
17 |
8 16
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝑦 ≠ ∅ ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐶 𝑦 ≠ ∅ ) |
19 |
|
mptexg |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
20 |
|
rnexg |
⊢ ( ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V → ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
21 |
1 19 20
|
3syl |
⊢ ( 𝜑 → ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
22 |
3 21
|
eqeltrid |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
23 |
|
raleq |
⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑦 ∈ 𝑐 𝑦 ≠ ∅ ↔ ∀ 𝑦 ∈ 𝐶 𝑦 ≠ ∅ ) ) |
24 |
|
id |
⊢ ( 𝑐 = 𝐶 → 𝑐 = 𝐶 ) |
25 |
|
unieq |
⊢ ( 𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶 ) |
26 |
24 25
|
feq23d |
⊢ ( 𝑐 = 𝐶 → ( 𝑓 : 𝑐 ⟶ ∪ 𝑐 ↔ 𝑓 : 𝐶 ⟶ ∪ 𝐶 ) ) |
27 |
|
raleq |
⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑦 ∈ 𝑐 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ↔ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) |
28 |
26 27
|
anbi12d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑓 : 𝑐 ⟶ ∪ 𝑐 ∧ ∀ 𝑦 ∈ 𝑐 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ↔ ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
29 |
28
|
exbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑓 ( 𝑓 : 𝑐 ⟶ ∪ 𝑐 ∧ ∀ 𝑦 ∈ 𝑐 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ↔ ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
30 |
23 29
|
imbi12d |
⊢ ( 𝑐 = 𝐶 → ( ( ∀ 𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃ 𝑓 ( 𝑓 : 𝑐 ⟶ ∪ 𝑐 ∧ ∀ 𝑦 ∈ 𝑐 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ) ) |
31 |
|
vex |
⊢ 𝑐 ∈ V |
32 |
31
|
ac5b |
⊢ ( ∀ 𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃ 𝑓 ( 𝑓 : 𝑐 ⟶ ∪ 𝑐 ∧ ∀ 𝑦 ∈ 𝑐 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) |
33 |
30 32
|
vtoclg |
⊢ ( 𝐶 ∈ V → ( ∀ 𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
34 |
22 33
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) ) |
35 |
18 34
|
mpd |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ) |
36 |
16
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) → ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 ) |
37 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) |
38 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
39 |
37 38
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) |
40 |
39
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) → ( 𝑦 = 𝐵 → ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) |
41 |
40
|
reximdva |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) → ( ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) |
42 |
36 41
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) → ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) |
43 |
42
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ( ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 → ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) |
44 |
43
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 → ∀ 𝑦 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) |
45 |
44
|
anim2d |
⊢ ( 𝜑 → ( ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) → ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) ) |
46 |
45
|
eximdv |
⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ( 𝑓 ‘ 𝑦 ) ∈ 𝑦 ) → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) ) |
47 |
35 46
|
mpd |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ ∪ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ 𝐵 ) ) |