Step |
Hyp |
Ref |
Expression |
1 |
|
cnmptcom.3 |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
cnmptcom.4 |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
3 |
|
cnmptcom.6 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐽 ×t 𝐾 ) Cn 𝐿 ) ) |
4 |
|
txtopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
5 |
1 2 4
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
6 |
|
cntop2 |
⊢ ( ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐽 ×t 𝐾 ) Cn 𝐿 ) → 𝐿 ∈ Top ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ Top ) |
8 |
|
toptopon2 |
⊢ ( 𝐿 ∈ Top ↔ 𝐿 ∈ ( TopOn ‘ ∪ 𝐿 ) ) |
9 |
7 8
|
sylib |
⊢ ( 𝜑 → 𝐿 ∈ ( TopOn ‘ ∪ 𝐿 ) ) |
10 |
|
cnf2 |
⊢ ( ( ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ∧ 𝐿 ∈ ( TopOn ‘ ∪ 𝐿 ) ∧ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) ∈ ( ( 𝐽 ×t 𝐾 ) Cn 𝐿 ) ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ∪ 𝐿 ) |
11 |
5 9 3 10
|
syl3anc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ∪ 𝐿 ) |
12 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) |
13 |
12
|
fmpo |
⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ∪ 𝐿 ) |
14 |
|
ralcom |
⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ ∀ 𝑦 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ) |
15 |
13 14
|
bitr3i |
⊢ ( ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) : ( 𝑋 × 𝑌 ) ⟶ ∪ 𝐿 ↔ ∀ 𝑦 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ) |
16 |
11 15
|
sylib |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ) |
17 |
|
eqid |
⊢ ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) |
18 |
17
|
fmpo |
⊢ ( ∀ 𝑦 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) : ( 𝑌 × 𝑋 ) ⟶ ∪ 𝐿 ) |
19 |
16 18
|
sylib |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) : ( 𝑌 × 𝑋 ) ⟶ ∪ 𝐿 ) |
20 |
19
|
ffnd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) Fn ( 𝑌 × 𝑋 ) ) |
21 |
|
fnov |
⊢ ( ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) Fn ( 𝑌 × 𝑋 ) ↔ ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
22 |
20 21
|
sylib |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
23 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑧 |
24 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑧 |
25 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑤 |
26 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
27 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑥 |
28 |
|
nfmpo2 |
⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) |
29 |
27 28 23
|
nfov |
⊢ Ⅎ 𝑦 ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) |
30 |
|
nfmpo1 |
⊢ Ⅎ 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) |
31 |
23 30 27
|
nfov |
⊢ Ⅎ 𝑦 ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) |
32 |
29 31
|
nfeq |
⊢ Ⅎ 𝑦 ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) |
33 |
26 32
|
nfim |
⊢ Ⅎ 𝑦 ( 𝜑 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) |
34 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
35 |
|
nfmpo1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) |
36 |
25 35 24
|
nfov |
⊢ Ⅎ 𝑥 ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) |
37 |
|
nfmpo2 |
⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) |
38 |
24 37 25
|
nfov |
⊢ Ⅎ 𝑥 ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) |
39 |
36 38
|
nfeq |
⊢ Ⅎ 𝑥 ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) |
40 |
34 39
|
nfim |
⊢ Ⅎ 𝑥 ( 𝜑 → ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) |
41 |
|
oveq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) ) |
42 |
|
oveq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) |
43 |
41 42
|
eqeq12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ↔ ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ) |
44 |
43
|
imbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝜑 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ↔ ( 𝜑 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ) ) |
45 |
|
oveq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) ) |
46 |
|
oveq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) |
47 |
45 46
|
eqeq12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ↔ ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
48 |
47
|
imbi2d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝜑 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ↔ ( 𝜑 → ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) ) |
49 |
|
rsp2 |
⊢ ( ∀ 𝑦 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 → ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ∪ 𝐿 ) ) |
50 |
49 16
|
syl11 |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ) → ( 𝜑 → 𝐴 ∈ ∪ 𝐿 ) ) |
51 |
12
|
ovmpt4g |
⊢ ( ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿 ) → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = 𝐴 ) |
52 |
51
|
3com12 |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿 ) → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = 𝐴 ) |
53 |
17
|
ovmpt4g |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿 ) → ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) = 𝐴 ) |
54 |
52 53
|
eqtr4d |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿 ) → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) |
55 |
54
|
3expia |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ) → ( 𝐴 ∈ ∪ 𝐿 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ) |
56 |
50 55
|
syld |
⊢ ( ( 𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ) → ( 𝜑 → ( 𝑥 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑦 ) = ( 𝑦 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑥 ) ) ) |
57 |
23 24 25 33 40 44 48 56
|
vtocl2gaf |
⊢ ( ( 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋 ) → ( 𝜑 → ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
58 |
57
|
com12 |
⊢ ( 𝜑 → ( ( 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋 ) → ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
59 |
58
|
3impib |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋 ) → ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) = ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) |
60 |
59
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) ) = ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑧 ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) 𝑤 ) ) ) |
61 |
22 60
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) ) ) |
62 |
2 1
|
cnmpt2nd |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ 𝑤 ) ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐽 ) ) |
63 |
2 1
|
cnmpt1st |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ 𝑧 ) ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐾 ) ) |
64 |
2 1 62 63 3
|
cnmpt22f |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝑌 , 𝑤 ∈ 𝑋 ↦ ( 𝑤 ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝐴 ) 𝑧 ) ) ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐿 ) ) |
65 |
61 64
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 , 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( ( 𝐾 ×t 𝐽 ) Cn 𝐿 ) ) |