Step |
Hyp |
Ref |
Expression |
1 |
|
ptunhmeo.x |
⊢ 𝑋 = ∪ 𝐾 |
2 |
|
ptunhmeo.y |
⊢ 𝑌 = ∪ 𝐿 |
3 |
|
ptunhmeo.j |
⊢ 𝐽 = ( ∏t ‘ 𝐹 ) |
4 |
|
ptunhmeo.k |
⊢ 𝐾 = ( ∏t ‘ ( 𝐹 ↾ 𝐴 ) ) |
5 |
|
ptunhmeo.l |
⊢ 𝐿 = ( ∏t ‘ ( 𝐹 ↾ 𝐵 ) ) |
6 |
|
ptunhmeo.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝑥 ∪ 𝑦 ) ) |
7 |
|
ptunhmeo.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
8 |
|
ptunhmeo.f |
⊢ ( 𝜑 → 𝐹 : 𝐶 ⟶ Top ) |
9 |
|
ptunhmeo.u |
⊢ ( 𝜑 → 𝐶 = ( 𝐴 ∪ 𝐵 ) ) |
10 |
|
ptunhmeo.i |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
11 |
|
vex |
⊢ 𝑥 ∈ V |
12 |
|
vex |
⊢ 𝑦 ∈ V |
13 |
11 12
|
op1std |
⊢ ( 𝑤 = 〈 𝑥 , 𝑦 〉 → ( 1st ‘ 𝑤 ) = 𝑥 ) |
14 |
11 12
|
op2ndd |
⊢ ( 𝑤 = 〈 𝑥 , 𝑦 〉 → ( 2nd ‘ 𝑤 ) = 𝑦 ) |
15 |
13 14
|
uneq12d |
⊢ ( 𝑤 = 〈 𝑥 , 𝑦 〉 → ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) = ( 𝑥 ∪ 𝑦 ) ) |
16 |
15
|
mpompt |
⊢ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝑥 ∪ 𝑦 ) ) |
17 |
6 16
|
eqtr4i |
⊢ 𝐺 = ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ) |
18 |
|
xp1st |
⊢ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) → ( 1st ‘ 𝑤 ) ∈ 𝑋 ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑤 ) ∈ 𝑋 ) |
20 |
|
ixpeq2 |
⊢ ( ∀ 𝑘 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) → X 𝑘 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
21 |
|
fvres |
⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
22 |
21
|
unieqd |
⊢ ( 𝑘 ∈ 𝐴 → ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) ) |
23 |
20 22
|
mprg |
⊢ X 𝑘 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) |
24 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) |
25 |
24 9
|
sseqtrrid |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
26 |
7 25
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
27 |
8 25
|
fssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) |
28 |
4
|
ptuni |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ∪ 𝐾 ) |
29 |
26 27 28
|
syl2anc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ∪ 𝐾 ) |
30 |
23 29
|
syl5eqr |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐾 ) |
31 |
30 1
|
eqtr4di |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑋 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑋 ) |
33 |
19 32
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑤 ) ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
34 |
|
xp2nd |
⊢ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) → ( 2nd ‘ 𝑤 ) ∈ 𝑌 ) |
35 |
34
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑤 ) ∈ 𝑌 ) |
36 |
9
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) = 𝐶 ) |
37 |
|
uneqdifeq |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐶 ↔ ( 𝐶 ∖ 𝐴 ) = 𝐵 ) ) |
38 |
25 10 37
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ∪ 𝐵 ) = 𝐶 ↔ ( 𝐶 ∖ 𝐴 ) = 𝐵 ) ) |
39 |
36 38
|
mpbid |
⊢ ( 𝜑 → ( 𝐶 ∖ 𝐴 ) = 𝐵 ) |
40 |
39
|
ixpeq1d |
⊢ ( 𝜑 → X 𝑘 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑘 ) = X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) ) |
41 |
|
ixpeq2 |
⊢ ( ∀ 𝑘 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) → X 𝑘 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) ) |
42 |
|
fvres |
⊢ ( 𝑘 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
43 |
42
|
unieqd |
⊢ ( 𝑘 ∈ 𝐵 → ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) ) |
44 |
41 43
|
mprg |
⊢ X 𝑘 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) |
45 |
|
ssun2 |
⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) |
46 |
45 9
|
sseqtrrid |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
47 |
7 46
|
ssexd |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
48 |
8 46
|
fssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) |
49 |
5
|
ptuni |
⊢ ( ( 𝐵 ∈ V ∧ ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) → X 𝑘 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ∪ 𝐿 ) |
50 |
47 48 49
|
syl2anc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ∪ 𝐿 ) |
51 |
44 50
|
syl5eqr |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐿 ) |
52 |
51 2
|
eqtr4di |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑌 ) |
53 |
40 52
|
eqtrd |
⊢ ( 𝜑 → X 𝑘 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑘 ) = 𝑌 ) |
54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑘 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑘 ) = 𝑌 ) |
55 |
35 54
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑤 ) ∈ X 𝑘 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑘 ) ) |
56 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → 𝐴 ⊆ 𝐶 ) |
57 |
|
undifixp |
⊢ ( ( ( 1st ‘ 𝑤 ) ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∧ ( 2nd ‘ 𝑤 ) ∈ X 𝑘 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑘 ) ∧ 𝐴 ⊆ 𝐶 ) → ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) |
58 |
33 55 56 57
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) |
59 |
3
|
ptuni |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 : 𝐶 ⟶ Top ) → X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐽 ) |
60 |
7 8 59
|
syl2anc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐽 ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐽 ) |
62 |
58 61
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ∈ ∪ 𝐽 ) |
63 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → 𝐴 ⊆ 𝐶 ) |
64 |
60
|
eleq2d |
⊢ ( 𝜑 → ( 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ↔ 𝑧 ∈ ∪ 𝐽 ) ) |
65 |
64
|
biimpar |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) |
66 |
|
resixp |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑧 ↾ 𝐴 ) ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
67 |
63 65 66
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → ( 𝑧 ↾ 𝐴 ) ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
68 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑋 ) |
69 |
67 68
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → ( 𝑧 ↾ 𝐴 ) ∈ 𝑋 ) |
70 |
46
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → 𝐵 ⊆ 𝐶 ) |
71 |
|
resixp |
⊢ ( ( 𝐵 ⊆ 𝐶 ∧ 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑧 ↾ 𝐵 ) ∈ X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) ) |
72 |
70 65 71
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → ( 𝑧 ↾ 𝐵 ) ∈ X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) ) |
73 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑌 ) |
74 |
72 73
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → ( 𝑧 ↾ 𝐵 ) ∈ 𝑌 ) |
75 |
69 74
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ∪ 𝐽 ) → 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ∈ ( 𝑋 × 𝑌 ) ) |
76 |
|
eqop |
⊢ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) → ( 𝑤 = 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ↔ ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) ) ) |
77 |
76
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝑤 = 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ↔ ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) ) ) |
78 |
65
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) ) |
79 |
|
ixpfn |
⊢ ( 𝑧 ∈ X 𝑘 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑘 ) → 𝑧 Fn 𝐶 ) |
80 |
|
fnresdm |
⊢ ( 𝑧 Fn 𝐶 → ( 𝑧 ↾ 𝐶 ) = 𝑧 ) |
81 |
78 79 80
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝑧 ↾ 𝐶 ) = 𝑧 ) |
82 |
9
|
reseq2d |
⊢ ( 𝜑 → ( 𝑧 ↾ 𝐶 ) = ( 𝑧 ↾ ( 𝐴 ∪ 𝐵 ) ) ) |
83 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝑧 ↾ 𝐶 ) = ( 𝑧 ↾ ( 𝐴 ∪ 𝐵 ) ) ) |
84 |
81 83
|
eqtr3d |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → 𝑧 = ( 𝑧 ↾ ( 𝐴 ∪ 𝐵 ) ) ) |
85 |
|
resundi |
⊢ ( 𝑧 ↾ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑧 ↾ 𝐴 ) ∪ ( 𝑧 ↾ 𝐵 ) ) |
86 |
84 85
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → 𝑧 = ( ( 𝑧 ↾ 𝐴 ) ∪ ( 𝑧 ↾ 𝐵 ) ) ) |
87 |
|
uneq12 |
⊢ ( ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) → ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) = ( ( 𝑧 ↾ 𝐴 ) ∪ ( 𝑧 ↾ 𝐵 ) ) ) |
88 |
87
|
eqeq2d |
⊢ ( ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) → ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↔ 𝑧 = ( ( 𝑧 ↾ 𝐴 ) ∪ ( 𝑧 ↾ 𝐵 ) ) ) ) |
89 |
86 88
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) → 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ) ) |
90 |
|
ixpfn |
⊢ ( ( 1st ‘ 𝑤 ) ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) → ( 1st ‘ 𝑤 ) Fn 𝐴 ) |
91 |
33 90
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑤 ) Fn 𝐴 ) |
92 |
91
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 1st ‘ 𝑤 ) Fn 𝐴 ) |
93 |
|
dffn2 |
⊢ ( ( 1st ‘ 𝑤 ) Fn 𝐴 ↔ ( 1st ‘ 𝑤 ) : 𝐴 ⟶ V ) |
94 |
92 93
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 1st ‘ 𝑤 ) : 𝐴 ⟶ V ) |
95 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) = 𝑌 ) |
96 |
35 95
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑤 ) ∈ X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) ) |
97 |
|
ixpfn |
⊢ ( ( 2nd ‘ 𝑤 ) ∈ X 𝑘 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑘 ) → ( 2nd ‘ 𝑤 ) Fn 𝐵 ) |
98 |
96 97
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑤 ) Fn 𝐵 ) |
99 |
98
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 2nd ‘ 𝑤 ) Fn 𝐵 ) |
100 |
|
dffn2 |
⊢ ( ( 2nd ‘ 𝑤 ) Fn 𝐵 ↔ ( 2nd ‘ 𝑤 ) : 𝐵 ⟶ V ) |
101 |
99 100
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 2nd ‘ 𝑤 ) : 𝐵 ⟶ V ) |
102 |
|
res0 |
⊢ ( ( 1st ‘ 𝑤 ) ↾ ∅ ) = ∅ |
103 |
|
res0 |
⊢ ( ( 2nd ‘ 𝑤 ) ↾ ∅ ) = ∅ |
104 |
102 103
|
eqtr4i |
⊢ ( ( 1st ‘ 𝑤 ) ↾ ∅ ) = ( ( 2nd ‘ 𝑤 ) ↾ ∅ ) |
105 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
106 |
105
|
reseq2d |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( 1st ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 1st ‘ 𝑤 ) ↾ ∅ ) ) |
107 |
105
|
reseq2d |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( 2nd ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 2nd ‘ 𝑤 ) ↾ ∅ ) ) |
108 |
104 106 107
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( 1st ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 2nd ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) ) |
109 |
|
fresaunres1 |
⊢ ( ( ( 1st ‘ 𝑤 ) : 𝐴 ⟶ V ∧ ( 2nd ‘ 𝑤 ) : 𝐵 ⟶ V ∧ ( ( 1st ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 2nd ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) = ( 1st ‘ 𝑤 ) ) |
110 |
94 101 108 109
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) = ( 1st ‘ 𝑤 ) ) |
111 |
110
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 1st ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) ) |
112 |
|
fresaunres2 |
⊢ ( ( ( 1st ‘ 𝑤 ) : 𝐴 ⟶ V ∧ ( 2nd ‘ 𝑤 ) : 𝐵 ⟶ V ∧ ( ( 1st ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 2nd ‘ 𝑤 ) ↾ ( 𝐴 ∩ 𝐵 ) ) ) → ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) = ( 2nd ‘ 𝑤 ) ) |
113 |
94 101 108 112
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) = ( 2nd ‘ 𝑤 ) ) |
114 |
113
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 2nd ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) ) |
115 |
111 114
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( 1st ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) ) ) |
116 |
|
reseq1 |
⊢ ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( 𝑧 ↾ 𝐴 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) ) |
117 |
116
|
eqeq2d |
⊢ ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ↔ ( 1st ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) ) ) |
118 |
|
reseq1 |
⊢ ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( 𝑧 ↾ 𝐵 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) ) |
119 |
118
|
eqeq2d |
⊢ ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ↔ ( 2nd ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) ) ) |
120 |
117 119
|
anbi12d |
⊢ ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) ↔ ( ( 1st ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ↾ 𝐵 ) ) ) ) |
121 |
115 120
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) → ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) ) ) |
122 |
89 121
|
impbid |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( ( ( 1st ‘ 𝑤 ) = ( 𝑧 ↾ 𝐴 ) ∧ ( 2nd ‘ 𝑤 ) = ( 𝑧 ↾ 𝐵 ) ) ↔ 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ) ) |
123 |
77 122
|
bitrd |
⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑧 ∈ ∪ 𝐽 ) ) → ( 𝑤 = 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ↔ 𝑧 = ( ( 1st ‘ 𝑤 ) ∪ ( 2nd ‘ 𝑤 ) ) ) ) |
124 |
17 62 75 123
|
f1ocnv2d |
⊢ ( 𝜑 → ( 𝐺 : ( 𝑋 × 𝑌 ) –1-1-onto→ ∪ 𝐽 ∧ ◡ 𝐺 = ( 𝑧 ∈ ∪ 𝐽 ↦ 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ) ) ) |
125 |
124
|
simprd |
⊢ ( 𝜑 → ◡ 𝐺 = ( 𝑧 ∈ ∪ 𝐽 ↦ 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ) ) |