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Theorem ptuncnv 21889
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 3352 . . . . . . 7 𝑥 ∈ V
3 vex 3352 . . . . . . 7 𝑦 ∈ V
42, 3op1std 7375 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
52, 3op2ndd 7376 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
64, 5uneq12d 3929 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → ((1st𝑤) ∪ (2nd𝑤)) = (𝑥𝑦))
76mpt2mpt 6949 . . . 4 (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2789 . . 3 𝐺 = (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤)))
9 xp1st 7397 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (1st𝑤) ∈ 𝑋)
109adantl 473 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ 𝑋)
11 ixpeq2 8126 . . . . . . . . . 10 (∀𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘))
12 fvres 6393 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1312unieqd 4603 . . . . . . . . . 10 (𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1411, 13mprg 3072 . . . . . . . . 9 X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘)
15 ptunhmeo.c . . . . . . . . . . 11 (𝜑𝐶𝑉)
16 ssun1 3937 . . . . . . . . . . . 12 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . 12 (𝜑𝐶 = (𝐴𝐵))
1816, 17syl5sseqr 3813 . . . . . . . . . . 11 (𝜑𝐴𝐶)
1915, 18ssexd 4965 . . . . . . . . . 10 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . 11 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6252 . . . . . . . . . 10 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . 11 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 21676 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2419, 21, 23syl2anc 579 . . . . . . . . 9 (𝜑X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2514, 24syl5eqr 2812 . . . . . . . 8 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝐾)
26 ptunhmeo.x . . . . . . . 8 𝑋 = 𝐾
2725, 26syl6eqr 2816 . . . . . . 7 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝑋)
2827adantr 472 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
2910, 28eleqtrrd 2846 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘))
30 xp2nd 7398 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (2nd𝑤) ∈ 𝑌)
3130adantl 473 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ 𝑌)
3217eqcomd 2770 . . . . . . . . . 10 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4216 . . . . . . . . . . 11 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 579 . . . . . . . . . 10 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 223 . . . . . . . . 9 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 8124 . . . . . . . 8 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = X𝑘𝐵 (𝐹𝑘))
38 ixpeq2 8126 . . . . . . . . . . 11 (∀𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘))
39 fvres 6393 . . . . . . . . . . . 12 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4039unieqd 4603 . . . . . . . . . . 11 (𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4138, 40mprg 3072 . . . . . . . . . 10 X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘)
42 ssun2 3938 . . . . . . . . . . . . 13 𝐵 ⊆ (𝐴𝐵)
4342, 17syl5sseqr 3813 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
4415, 43ssexd 4965 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
4520, 43fssresd 6252 . . . . . . . . . . 11 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . 12 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 21676 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4844, 45, 47syl2anc 579 . . . . . . . . . 10 (𝜑X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4941, 48syl5eqr 2812 . . . . . . . . 9 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝐿)
50 ptunhmeo.y . . . . . . . . 9 𝑌 = 𝐿
5149, 50syl6eqr 2816 . . . . . . . 8 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝑌)
5237, 51eqtrd 2798 . . . . . . 7 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5352adantr 472 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5431, 53eleqtrrd 2846 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘))
5518adantr 472 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 8148 . . . . 5 (((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) ∧ (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) ∧ 𝐴𝐶) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
5729, 54, 55, 56syl3anc 1490 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
58 ptunhmeo.j . . . . . . 7 𝐽 = (∏t𝐹)
5958ptuni 21676 . . . . . 6 ((𝐶𝑉𝐹:𝐶⟶Top) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6015, 20, 59syl2anc 579 . . . . 5 (𝜑X𝑘𝐶 (𝐹𝑘) = 𝐽)
6160adantr 472 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6257, 61eleqtrd 2845 . . 3 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝐽)
6318adantr 472 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐴𝐶)
6460eleq2d 2829 . . . . . . 7 (𝜑 → (𝑧X𝑘𝐶 (𝐹𝑘) ↔ 𝑧 𝐽))
6564biimpar 469 . . . . . 6 ((𝜑𝑧 𝐽) → 𝑧X𝑘𝐶 (𝐹𝑘))
66 resixp 8147 . . . . . 6 ((𝐴𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6763, 65, 66syl2anc 579 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6827adantr 472 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
6967, 68eleqtrd 2845 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ 𝑋)
7043adantr 472 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐵𝐶)
71 resixp 8147 . . . . . 6 ((𝐵𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7270, 65, 71syl2anc 579 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7351adantr 472 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
7472, 73eleqtrd 2845 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ 𝑌)
75 opelxpi 5313 . . . 4 (((𝑧𝐴) ∈ 𝑋 ∧ (𝑧𝐵) ∈ 𝑌) → ⟨(𝑧𝐴), (𝑧𝐵)⟩ ∈ (𝑋 × 𝑌))
7669, 74, 75syl2anc 579 . . 3 ((𝜑𝑧 𝐽) → ⟨(𝑧𝐴), (𝑧𝐵)⟩ ∈ (𝑋 × 𝑌))
77 eqop 7407 . . . . 5 (𝑤 ∈ (𝑋 × 𝑌) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7877ad2antrl 719 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7965adantrl 707 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧X𝑘𝐶 (𝐹𝑘))
80 ixpfn 8118 . . . . . . . . 9 (𝑧X𝑘𝐶 (𝐹𝑘) → 𝑧 Fn 𝐶)
81 fnresdm 6177 . . . . . . . . 9 (𝑧 Fn 𝐶 → (𝑧𝐶) = 𝑧)
8279, 80, 813syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = 𝑧)
8317reseq2d 5564 . . . . . . . . 9 (𝜑 → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8483adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8582, 84eqtr3d 2800 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = (𝑧 ↾ (𝐴𝐵)))
86 resundi 5585 . . . . . . 7 (𝑧 ↾ (𝐴𝐵)) = ((𝑧𝐴) ∪ (𝑧𝐵))
8785, 86syl6eq 2814 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵)))
88 uneq12 3923 . . . . . . 7 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → ((1st𝑤) ∪ (2nd𝑤)) = ((𝑧𝐴) ∪ (𝑧𝐵)))
8988eqeq2d 2774 . . . . . 6 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) ↔ 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵))))
9087, 89syl5ibrcom 238 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
91 ixpfn 8118 . . . . . . . . . . . 12 ((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) → (1st𝑤) Fn 𝐴)
9229, 91syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) Fn 𝐴)
9392adantrr 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) Fn 𝐴)
94 dffn2 6224 . . . . . . . . . 10 ((1st𝑤) Fn 𝐴 ↔ (1st𝑤):𝐴⟶V)
9593, 94sylib 209 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤):𝐴⟶V)
9651adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
9731, 96eleqtrrd 2846 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘))
98 ixpfn 8118 . . . . . . . . . . . 12 ((2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘) → (2nd𝑤) Fn 𝐵)
9997, 98syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) Fn 𝐵)
10099adantrr 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) Fn 𝐵)
101 dffn2 6224 . . . . . . . . . 10 ((2nd𝑤) Fn 𝐵 ↔ (2nd𝑤):𝐵⟶V)
102100, 101sylib 209 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤):𝐵⟶V)
103 res0 5568 . . . . . . . . . . 11 ((1st𝑤) ↾ ∅) = ∅
104 res0 5568 . . . . . . . . . . 11 ((2nd𝑤) ↾ ∅) = ∅
105103, 104eqtr4i 2789 . . . . . . . . . 10 ((1st𝑤) ↾ ∅) = ((2nd𝑤) ↾ ∅)
10633adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝐴𝐵) = ∅)
107106reseq2d 5564 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((1st𝑤) ↾ ∅))
108106reseq2d 5564 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((2nd𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ ∅))
109105, 107, 1083eqtr4a 2824 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵)))
110 fresaunres1 6258 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
11195, 102, 109, 110syl3anc 1490 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
112111eqcomd 2770 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
113 fresaunres2 6257 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
11495, 102, 109, 113syl3anc 1490 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
115114eqcomd 2770 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
116112, 115jca 507 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
117 reseq1 5558 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐴) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
118117eqeq2d 2774 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ↔ (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴)))
119 reseq1 5558 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐵) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
120119eqeq2d 2774 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((2nd𝑤) = (𝑧𝐵) ↔ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
121118, 120anbi12d 624 . . . . . 6 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))))
122116, 121syl5ibrcom 238 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
12390, 122impbid 203 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
12478, 123bitrd 270 . . 3 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
1258, 62, 76, 124f1ocnv2d 7083 . 2 (𝜑 → (𝐺:(𝑋 × 𝑌)–1-1-onto 𝐽𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩)))
126125simprd 489 1 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  Vcvv 3349  cdif 3728  cun 3729  cin 3730  wss 3731  c0 4078  cop 4339   cuni 4593  cmpt 4887   × cxp 5274  ccnv 5275  cres 5278   Fn wfn 6062  wf 6063  1-1-ontowf1o 6066  cfv 6067  cmpt2 6843  1st c1st 7363  2nd c2nd 7364  Xcixp 8112  tcpt 16366  Topctop 20976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-ixp 8113  df-en 8160  df-fin 8163  df-fi 8523  df-topgen 16371  df-pt 16372  df-top 20977  df-bases 21029
This theorem is referenced by:  ptunhmeo  21890
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