Step |
Hyp |
Ref |
Expression |
1 |
|
ixpfi2.1 |
⊢ ( 𝜑 → 𝐶 ∈ Fin ) |
2 |
|
ixpfi2.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ Fin ) |
3 |
|
ixpfi2.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → 𝐵 ⊆ { 𝐷 } ) |
4 |
|
inss2 |
⊢ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 |
5 |
|
ssfi |
⊢ ( ( 𝐶 ∈ Fin ∧ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐶 ) → ( 𝐴 ∩ 𝐶 ) ∈ Fin ) |
6 |
1 4 5
|
sylancl |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ∈ Fin ) |
7 |
|
inss1 |
⊢ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 |
8 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ Fin ) |
9 |
|
ssralv |
⊢ ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ) ) |
10 |
7 8 9
|
mpsyl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ) |
11 |
|
ixpfi |
⊢ ( ( ( 𝐴 ∩ 𝐶 ) ∈ Fin ∧ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ) → X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ) |
12 |
6 10 11
|
syl2anc |
⊢ ( 𝜑 → X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ) |
13 |
|
resixp |
⊢ ( ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ∧ 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ) → ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ∈ X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ) |
14 |
7 13
|
mpan |
⊢ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 → ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ∈ X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ) |
15 |
14
|
a1i |
⊢ ( 𝜑 → ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 → ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ∈ X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ) ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ) |
17 |
|
vex |
⊢ 𝑓 ∈ V |
18 |
17
|
elixp |
⊢ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
19 |
16 18
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑓 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) ) |
20 |
19
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
21 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) |
22 |
|
vex |
⊢ 𝑔 ∈ V |
23 |
22
|
elixp |
⊢ ( 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ) |
24 |
21 23
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑔 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ) |
25 |
24
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) |
26 |
|
r19.26 |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ) |
27 |
|
difss |
⊢ ( 𝐴 ∖ 𝐶 ) ⊆ 𝐴 |
28 |
|
ssralv |
⊢ ( ( 𝐴 ∖ 𝐶 ) ⊆ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ) ) |
29 |
27 28
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) ) |
30 |
3
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑓 ‘ 𝑥 ) ∈ { 𝐷 } ) ) |
31 |
|
elsni |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ { 𝐷 } → ( 𝑓 ‘ 𝑥 ) = 𝐷 ) |
32 |
30 31
|
syl6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑓 ‘ 𝑥 ) = 𝐷 ) ) |
33 |
3
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑔 ‘ 𝑥 ) ∈ { 𝐷 } ) ) |
34 |
|
elsni |
⊢ ( ( 𝑔 ‘ 𝑥 ) ∈ { 𝐷 } → ( 𝑔 ‘ 𝑥 ) = 𝐷 ) |
35 |
33 34
|
syl6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑔 ‘ 𝑥 ) = 𝐷 ) ) |
36 |
32 35
|
anim12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ( ( 𝑓 ‘ 𝑥 ) = 𝐷 ∧ ( 𝑔 ‘ 𝑥 ) = 𝐷 ) ) ) |
37 |
|
eqtr3 |
⊢ ( ( ( 𝑓 ‘ 𝑥 ) = 𝐷 ∧ ( 𝑔 ‘ 𝑥 ) = 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) |
38 |
36 37
|
syl6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
39 |
38
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
41 |
29 40
|
syl5 |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
42 |
26 41
|
syl5bir |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
43 |
20 25 42
|
mp2and |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) |
44 |
43
|
biantrud |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) ) |
45 |
|
fvres |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( 𝑓 ‘ 𝑥 ) ) |
46 |
|
fvres |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) |
47 |
45 46
|
eqeq12d |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) → ( ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) ↔ ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
48 |
47
|
ralbiia |
⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) |
49 |
|
inundif |
⊢ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐴 ∖ 𝐶 ) ) = 𝐴 |
50 |
49
|
raleqi |
⊢ ( ∀ 𝑥 ∈ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐴 ∖ 𝐶 ) ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) |
51 |
|
ralunb |
⊢ ( ∀ 𝑥 ∈ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐴 ∖ 𝐶 ) ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
52 |
50 51
|
bitr3i |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ↔ ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ∧ ∀ 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
53 |
44 48 52
|
3bitr4g |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
54 |
19
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑓 Fn 𝐴 ) |
55 |
|
fnssres |
⊢ ( ( 𝑓 Fn 𝐴 ∧ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ) → ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ) |
56 |
54 7 55
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ) |
57 |
24
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑔 Fn 𝐴 ) |
58 |
|
fnssres |
⊢ ( ( 𝑔 Fn 𝐴 ∧ ( 𝐴 ∩ 𝐶 ) ⊆ 𝐴 ) → ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ) |
59 |
57 7 58
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ) |
60 |
|
eqfnfv |
⊢ ( ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ∧ ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) Fn ( 𝐴 ∩ 𝐶 ) ) → ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) = ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) ) ) |
61 |
56 59 60
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) = ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) = ( ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ‘ 𝑥 ) ) ) |
62 |
|
eqfnfv |
⊢ ( ( 𝑓 Fn 𝐴 ∧ 𝑔 Fn 𝐴 ) → ( 𝑓 = 𝑔 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
63 |
54 57 62
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑓 = 𝑔 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑓 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑥 ) ) ) |
64 |
53 61 63
|
3bitr4d |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) = ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ↔ 𝑓 = 𝑔 ) ) |
65 |
64
|
ex |
⊢ ( 𝜑 → ( ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝑔 ∈ X 𝑥 ∈ 𝐴 𝐵 ) → ( ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) = ( 𝑔 ↾ ( 𝐴 ∩ 𝐶 ) ) ↔ 𝑓 = 𝑔 ) ) ) |
66 |
15 65
|
dom2lem |
⊢ ( 𝜑 → ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ↦ ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ) : X 𝑥 ∈ 𝐴 𝐵 –1-1→ X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ) |
67 |
|
f1fi |
⊢ ( ( X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ∈ Fin ∧ ( 𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵 ↦ ( 𝑓 ↾ ( 𝐴 ∩ 𝐶 ) ) ) : X 𝑥 ∈ 𝐴 𝐵 –1-1→ X 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) 𝐵 ) → X 𝑥 ∈ 𝐴 𝐵 ∈ Fin ) |
68 |
12 66 67
|
syl2anc |
⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐵 ∈ Fin ) |