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Theorem ptuncnv 22345
Description: Exhibit the converse function of the map 𝐺 which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
ptunhmeo.x 𝑋 = 𝐾
ptunhmeo.y 𝑌 = 𝐿
ptunhmeo.j 𝐽 = (∏t𝐹)
ptunhmeo.k 𝐾 = (∏t‘(𝐹𝐴))
ptunhmeo.l 𝐿 = (∏t‘(𝐹𝐵))
ptunhmeo.g 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
ptunhmeo.c (𝜑𝐶𝑉)
ptunhmeo.f (𝜑𝐹:𝐶⟶Top)
ptunhmeo.u (𝜑𝐶 = (𝐴𝐵))
ptunhmeo.i (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
ptuncnv (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑧,𝐺   𝜑,𝑥,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑧,𝑉   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ptuncnv
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptunhmeo.g . . . 4 𝐺 = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
2 vex 3498 . . . . . . 7 𝑥 ∈ V
3 vex 3498 . . . . . . 7 𝑦 ∈ V
42, 3op1std 7690 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
52, 3op2ndd 7691 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
64, 5uneq12d 4139 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → ((1st𝑤) ∪ (2nd𝑤)) = (𝑥𝑦))
76mpompt 7255 . . . 4 (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2847 . . 3 𝐺 = (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤)))
9 xp1st 7712 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (1st𝑤) ∈ 𝑋)
109adantl 482 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ 𝑋)
11 ixpeq2 8464 . . . . . . . . . 10 (∀𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘))
12 fvres 6683 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1312unieqd 4842 . . . . . . . . . 10 (𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1411, 13mprg 3152 . . . . . . . . 9 X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘)
15 ptunhmeo.c . . . . . . . . . . 11 (𝜑𝐶𝑉)
16 ssun1 4147 . . . . . . . . . . . 12 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . 12 (𝜑𝐶 = (𝐴𝐵))
1816, 17sseqtrrid 4019 . . . . . . . . . . 11 (𝜑𝐴𝐶)
1915, 18ssexd 5220 . . . . . . . . . 10 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . 11 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6539 . . . . . . . . . 10 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . 11 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 22132 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2419, 21, 23syl2anc 584 . . . . . . . . 9 (𝜑X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2514, 24syl5eqr 2870 . . . . . . . 8 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝐾)
26 ptunhmeo.x . . . . . . . 8 𝑋 = 𝐾
2725, 26syl6eqr 2874 . . . . . . 7 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝑋)
2827adantr 481 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
2910, 28eleqtrrd 2916 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘))
30 xp2nd 7713 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (2nd𝑤) ∈ 𝑌)
3130adantl 482 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ 𝑌)
3217eqcomd 2827 . . . . . . . . . 10 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4436 . . . . . . . . . . 11 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 233 . . . . . . . . 9 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 8462 . . . . . . . 8 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = X𝑘𝐵 (𝐹𝑘))
38 ixpeq2 8464 . . . . . . . . . . 11 (∀𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘))
39 fvres 6683 . . . . . . . . . . . 12 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4039unieqd 4842 . . . . . . . . . . 11 (𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4138, 40mprg 3152 . . . . . . . . . 10 X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘)
42 ssun2 4148 . . . . . . . . . . . . 13 𝐵 ⊆ (𝐴𝐵)
4342, 17sseqtrrid 4019 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
4415, 43ssexd 5220 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
4520, 43fssresd 6539 . . . . . . . . . . 11 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . 12 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 22132 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4844, 45, 47syl2anc 584 . . . . . . . . . 10 (𝜑X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4941, 48syl5eqr 2870 . . . . . . . . 9 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝐿)
50 ptunhmeo.y . . . . . . . . 9 𝑌 = 𝐿
5149, 50syl6eqr 2874 . . . . . . . 8 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝑌)
5237, 51eqtrd 2856 . . . . . . 7 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5352adantr 481 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5431, 53eleqtrrd 2916 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘))
5518adantr 481 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 8487 . . . . 5 (((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) ∧ (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) ∧ 𝐴𝐶) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
5729, 54, 55, 56syl3anc 1363 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
58 ptunhmeo.j . . . . . . 7 𝐽 = (∏t𝐹)
5958ptuni 22132 . . . . . 6 ((𝐶𝑉𝐹:𝐶⟶Top) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6015, 20, 59syl2anc 584 . . . . 5 (𝜑X𝑘𝐶 (𝐹𝑘) = 𝐽)
6160adantr 481 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6257, 61eleqtrd 2915 . . 3 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝐽)
6318adantr 481 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐴𝐶)
6460eleq2d 2898 . . . . . . 7 (𝜑 → (𝑧X𝑘𝐶 (𝐹𝑘) ↔ 𝑧 𝐽))
6564biimpar 478 . . . . . 6 ((𝜑𝑧 𝐽) → 𝑧X𝑘𝐶 (𝐹𝑘))
66 resixp 8486 . . . . . 6 ((𝐴𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6763, 65, 66syl2anc 584 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6827adantr 481 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
6967, 68eleqtrd 2915 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ 𝑋)
7043adantr 481 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐵𝐶)
71 resixp 8486 . . . . . 6 ((𝐵𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7270, 65, 71syl2anc 584 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7351adantr 481 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
7472, 73eleqtrd 2915 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ 𝑌)
7569, 74opelxpd 5587 . . 3 ((𝜑𝑧 𝐽) → ⟨(𝑧𝐴), (𝑧𝐵)⟩ ∈ (𝑋 × 𝑌))
76 eqop 7722 . . . . 5 (𝑤 ∈ (𝑋 × 𝑌) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7776ad2antrl 724 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7865adantrl 712 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧X𝑘𝐶 (𝐹𝑘))
79 ixpfn 8456 . . . . . . . . 9 (𝑧X𝑘𝐶 (𝐹𝑘) → 𝑧 Fn 𝐶)
80 fnresdm 6460 . . . . . . . . 9 (𝑧 Fn 𝐶 → (𝑧𝐶) = 𝑧)
8178, 79, 803syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = 𝑧)
8217reseq2d 5847 . . . . . . . . 9 (𝜑 → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8382adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8481, 83eqtr3d 2858 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = (𝑧 ↾ (𝐴𝐵)))
85 resundi 5861 . . . . . . 7 (𝑧 ↾ (𝐴𝐵)) = ((𝑧𝐴) ∪ (𝑧𝐵))
8684, 85syl6eq 2872 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵)))
87 uneq12 4133 . . . . . . 7 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → ((1st𝑤) ∪ (2nd𝑤)) = ((𝑧𝐴) ∪ (𝑧𝐵)))
8887eqeq2d 2832 . . . . . 6 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) ↔ 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵))))
8986, 88syl5ibrcom 248 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
90 ixpfn 8456 . . . . . . . . . . . 12 ((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) → (1st𝑤) Fn 𝐴)
9129, 90syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) Fn 𝐴)
9291adantrr 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) Fn 𝐴)
93 dffn2 6510 . . . . . . . . . 10 ((1st𝑤) Fn 𝐴 ↔ (1st𝑤):𝐴⟶V)
9492, 93sylib 219 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤):𝐴⟶V)
9551adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
9631, 95eleqtrrd 2916 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘))
97 ixpfn 8456 . . . . . . . . . . . 12 ((2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘) → (2nd𝑤) Fn 𝐵)
9896, 97syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) Fn 𝐵)
9998adantrr 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) Fn 𝐵)
100 dffn2 6510 . . . . . . . . . 10 ((2nd𝑤) Fn 𝐵 ↔ (2nd𝑤):𝐵⟶V)
10199, 100sylib 219 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤):𝐵⟶V)
102 res0 5851 . . . . . . . . . . 11 ((1st𝑤) ↾ ∅) = ∅
103 res0 5851 . . . . . . . . . . 11 ((2nd𝑤) ↾ ∅) = ∅
104102, 103eqtr4i 2847 . . . . . . . . . 10 ((1st𝑤) ↾ ∅) = ((2nd𝑤) ↾ ∅)
10533adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝐴𝐵) = ∅)
106105reseq2d 5847 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((1st𝑤) ↾ ∅))
107105reseq2d 5847 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((2nd𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ ∅))
108104, 106, 1073eqtr4a 2882 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵)))
109 fresaunres1 6545 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
11094, 101, 108, 109syl3anc 1363 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
111110eqcomd 2827 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
112 fresaunres2 6544 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
11394, 101, 108, 112syl3anc 1363 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
114113eqcomd 2827 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
115111, 114jca 512 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
116 reseq1 5841 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐴) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
117116eqeq2d 2832 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ↔ (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴)))
118 reseq1 5841 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐵) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
119118eqeq2d 2832 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((2nd𝑤) = (𝑧𝐵) ↔ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
120117, 119anbi12d 630 . . . . . 6 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))))
121115, 120syl5ibrcom 248 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
12289, 121impbid 213 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
12377, 122bitrd 280 . . 3 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
1248, 62, 75, 123f1ocnv2d 7387 . 2 (𝜑 → (𝐺:(𝑋 × 𝑌)–1-1-onto 𝐽𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩)))
125124simprd 496 1 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3495  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  cop 4565   cuni 4832  cmpt 5138   × cxp 5547  ccnv 5548  cres 5551   Fn wfn 6344  wf 6345  1-1-ontowf1o 6348  cfv 6349  cmpo 7147  1st c1st 7678  2nd c2nd 7679  Xcixp 8450  tcpt 16702  Topctop 21431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-ixp 8451  df-en 8499  df-fin 8502  df-fi 8864  df-topgen 16707  df-pt 16708  df-top 21432  df-bases 21484
This theorem is referenced by:  ptunhmeo  22346
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