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Theorem ptuncnv 21658
 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 3234 . . . . . . 7 𝑥 ∈ V
3 vex 3234 . . . . . . 7 𝑦 ∈ V
42, 3op1std 7220 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) = 𝑥)
52, 3op2ndd 7221 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd𝑤) = 𝑦)
64, 5uneq12d 3801 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → ((1st𝑤) ∪ (2nd𝑤)) = (𝑥𝑦))
76mpt2mpt 6794 . . . 4 (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤))) = (𝑥𝑋, 𝑦𝑌 ↦ (𝑥𝑦))
81, 7eqtr4i 2676 . . 3 𝐺 = (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st𝑤) ∪ (2nd𝑤)))
9 xp1st 7242 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (1st𝑤) ∈ 𝑋)
109adantl 481 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ 𝑋)
11 ixpeq2 7964 . . . . . . . . . 10 (∀𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘))
12 fvres 6245 . . . . . . . . . . 11 (𝑘𝐴 → ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1312unieqd 4478 . . . . . . . . . 10 (𝑘𝐴 ((𝐹𝐴)‘𝑘) = (𝐹𝑘))
1411, 13mprg 2955 . . . . . . . . 9 X𝑘𝐴 ((𝐹𝐴)‘𝑘) = X𝑘𝐴 (𝐹𝑘)
15 ptunhmeo.c . . . . . . . . . . 11 (𝜑𝐶𝑉)
16 ssun1 3809 . . . . . . . . . . . 12 𝐴 ⊆ (𝐴𝐵)
17 ptunhmeo.u . . . . . . . . . . . 12 (𝜑𝐶 = (𝐴𝐵))
1816, 17syl5sseqr 3687 . . . . . . . . . . 11 (𝜑𝐴𝐶)
1915, 18ssexd 4838 . . . . . . . . . 10 (𝜑𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . 11 (𝜑𝐹:𝐶⟶Top)
2120, 18fssresd 6109 . . . . . . . . . 10 (𝜑 → (𝐹𝐴):𝐴⟶Top)
22 ptunhmeo.k . . . . . . . . . . 11 𝐾 = (∏t‘(𝐹𝐴))
2322ptuni 21445 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹𝐴):𝐴⟶Top) → X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2419, 21, 23syl2anc 694 . . . . . . . . 9 (𝜑X𝑘𝐴 ((𝐹𝐴)‘𝑘) = 𝐾)
2514, 24syl5eqr 2699 . . . . . . . 8 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝐾)
26 ptunhmeo.x . . . . . . . 8 𝑋 = 𝐾
2725, 26syl6eqr 2703 . . . . . . 7 (𝜑X𝑘𝐴 (𝐹𝑘) = 𝑋)
2827adantr 480 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
2910, 28eleqtrrd 2733 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘))
30 xp2nd 7243 . . . . . . 7 (𝑤 ∈ (𝑋 × 𝑌) → (2nd𝑤) ∈ 𝑌)
3130adantl 481 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ 𝑌)
3217eqcomd 2657 . . . . . . . . . 10 (𝜑 → (𝐴𝐵) = 𝐶)
33 ptunhmeo.i . . . . . . . . . . 11 (𝜑 → (𝐴𝐵) = ∅)
34 uneqdifeq 4090 . . . . . . . . . . 11 ((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3518, 33, 34syl2anc 694 . . . . . . . . . 10 (𝜑 → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))
3632, 35mpbid 222 . . . . . . . . 9 (𝜑 → (𝐶𝐴) = 𝐵)
3736ixpeq1d 7962 . . . . . . . 8 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = X𝑘𝐵 (𝐹𝑘))
38 ixpeq2 7964 . . . . . . . . . . 11 (∀𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘))
39 fvres 6245 . . . . . . . . . . . 12 (𝑘𝐵 → ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4039unieqd 4478 . . . . . . . . . . 11 (𝑘𝐵 ((𝐹𝐵)‘𝑘) = (𝐹𝑘))
4138, 40mprg 2955 . . . . . . . . . 10 X𝑘𝐵 ((𝐹𝐵)‘𝑘) = X𝑘𝐵 (𝐹𝑘)
42 ssun2 3810 . . . . . . . . . . . . 13 𝐵 ⊆ (𝐴𝐵)
4342, 17syl5sseqr 3687 . . . . . . . . . . . 12 (𝜑𝐵𝐶)
4415, 43ssexd 4838 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
4520, 43fssresd 6109 . . . . . . . . . . 11 (𝜑 → (𝐹𝐵):𝐵⟶Top)
46 ptunhmeo.l . . . . . . . . . . . 12 𝐿 = (∏t‘(𝐹𝐵))
4746ptuni 21445 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝐹𝐵):𝐵⟶Top) → X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4844, 45, 47syl2anc 694 . . . . . . . . . 10 (𝜑X𝑘𝐵 ((𝐹𝐵)‘𝑘) = 𝐿)
4941, 48syl5eqr 2699 . . . . . . . . 9 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝐿)
50 ptunhmeo.y . . . . . . . . 9 𝑌 = 𝐿
5149, 50syl6eqr 2703 . . . . . . . 8 (𝜑X𝑘𝐵 (𝐹𝑘) = 𝑌)
5237, 51eqtrd 2685 . . . . . . 7 (𝜑X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5352adantr 480 . . . . . 6 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) = 𝑌)
5431, 53eleqtrrd 2733 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘))
5518adantr 480 . . . . 5 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → 𝐴𝐶)
56 undifixp 7986 . . . . 5 (((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) ∧ (2nd𝑤) ∈ X𝑘 ∈ (𝐶𝐴) (𝐹𝑘) ∧ 𝐴𝐶) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
5729, 54, 55, 56syl3anc 1366 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ X𝑘𝐶 (𝐹𝑘))
58 ptunhmeo.j . . . . . . 7 𝐽 = (∏t𝐹)
5958ptuni 21445 . . . . . 6 ((𝐶𝑉𝐹:𝐶⟶Top) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6015, 20, 59syl2anc 694 . . . . 5 (𝜑X𝑘𝐶 (𝐹𝑘) = 𝐽)
6160adantr 480 . . . 4 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐶 (𝐹𝑘) = 𝐽)
6257, 61eleqtrd 2732 . . 3 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝐽)
6318adantr 480 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐴𝐶)
6460eleq2d 2716 . . . . . . 7 (𝜑 → (𝑧X𝑘𝐶 (𝐹𝑘) ↔ 𝑧 𝐽))
6564biimpar 501 . . . . . 6 ((𝜑𝑧 𝐽) → 𝑧X𝑘𝐶 (𝐹𝑘))
66 resixp 7985 . . . . . 6 ((𝐴𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6763, 65, 66syl2anc 694 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ X𝑘𝐴 (𝐹𝑘))
6827adantr 480 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐴 (𝐹𝑘) = 𝑋)
6967, 68eleqtrd 2732 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐴) ∈ 𝑋)
7043adantr 480 . . . . . 6 ((𝜑𝑧 𝐽) → 𝐵𝐶)
71 resixp 7985 . . . . . 6 ((𝐵𝐶𝑧X𝑘𝐶 (𝐹𝑘)) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7270, 65, 71syl2anc 694 . . . . 5 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ X𝑘𝐵 (𝐹𝑘))
7351adantr 480 . . . . 5 ((𝜑𝑧 𝐽) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
7472, 73eleqtrd 2732 . . . 4 ((𝜑𝑧 𝐽) → (𝑧𝐵) ∈ 𝑌)
75 opelxpi 5182 . . . 4 (((𝑧𝐴) ∈ 𝑋 ∧ (𝑧𝐵) ∈ 𝑌) → ⟨(𝑧𝐴), (𝑧𝐵)⟩ ∈ (𝑋 × 𝑌))
7669, 74, 75syl2anc 694 . . 3 ((𝜑𝑧 𝐽) → ⟨(𝑧𝐴), (𝑧𝐵)⟩ ∈ (𝑋 × 𝑌))
77 eqop 7252 . . . . 5 (𝑤 ∈ (𝑋 × 𝑌) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7877ad2antrl 764 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
7965adantrl 752 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧X𝑘𝐶 (𝐹𝑘))
80 ixpfn 7956 . . . . . . . . 9 (𝑧X𝑘𝐶 (𝐹𝑘) → 𝑧 Fn 𝐶)
81 fnresdm 6038 . . . . . . . . 9 (𝑧 Fn 𝐶 → (𝑧𝐶) = 𝑧)
8279, 80, 813syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = 𝑧)
8317reseq2d 5428 . . . . . . . . 9 (𝜑 → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8483adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧𝐶) = (𝑧 ↾ (𝐴𝐵)))
8582, 84eqtr3d 2687 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = (𝑧 ↾ (𝐴𝐵)))
86 resundi 5445 . . . . . . 7 (𝑧 ↾ (𝐴𝐵)) = ((𝑧𝐴) ∪ (𝑧𝐵))
8785, 86syl6eq 2701 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵)))
88 uneq12 3795 . . . . . . 7 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → ((1st𝑤) ∪ (2nd𝑤)) = ((𝑧𝐴) ∪ (𝑧𝐵)))
8988eqeq2d 2661 . . . . . 6 (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) ↔ 𝑧 = ((𝑧𝐴) ∪ (𝑧𝐵))))
9087, 89syl5ibrcom 237 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) → 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
91 ixpfn 7956 . . . . . . . . . . . 12 ((1st𝑤) ∈ X𝑘𝐴 (𝐹𝑘) → (1st𝑤) Fn 𝐴)
9229, 91syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (1st𝑤) Fn 𝐴)
9392adantrr 753 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) Fn 𝐴)
94 dffn2 6085 . . . . . . . . . 10 ((1st𝑤) Fn 𝐴 ↔ (1st𝑤):𝐴⟶V)
9593, 94sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤):𝐴⟶V)
9651adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → X𝑘𝐵 (𝐹𝑘) = 𝑌)
9731, 96eleqtrrd 2733 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘))
98 ixpfn 7956 . . . . . . . . . . . 12 ((2nd𝑤) ∈ X𝑘𝐵 (𝐹𝑘) → (2nd𝑤) Fn 𝐵)
9997, 98syl 17 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑋 × 𝑌)) → (2nd𝑤) Fn 𝐵)
10099adantrr 753 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) Fn 𝐵)
101 dffn2 6085 . . . . . . . . . 10 ((2nd𝑤) Fn 𝐵 ↔ (2nd𝑤):𝐵⟶V)
102100, 101sylib 208 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤):𝐵⟶V)
103 res0 5432 . . . . . . . . . . 11 ((1st𝑤) ↾ ∅) = ∅
104 res0 5432 . . . . . . . . . . 11 ((2nd𝑤) ↾ ∅) = ∅
105103, 104eqtr4i 2676 . . . . . . . . . 10 ((1st𝑤) ↾ ∅) = ((2nd𝑤) ↾ ∅)
10633adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝐴𝐵) = ∅)
107106reseq2d 5428 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((1st𝑤) ↾ ∅))
108106reseq2d 5428 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((2nd𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ ∅))
109105, 107, 1083eqtr4a 2711 . . . . . . . . 9 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵)))
110 fresaunres1 6115 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
11195, 102, 109, 110syl3anc 1366 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) = (1st𝑤))
112111eqcomd 2657 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
113 fresaunres2 6114 . . . . . . . . 9 (((1st𝑤):𝐴⟶V ∧ (2nd𝑤):𝐵⟶V ∧ ((1st𝑤) ↾ (𝐴𝐵)) = ((2nd𝑤) ↾ (𝐴𝐵))) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
11495, 102, 109, 113syl3anc 1366 . . . . . . . 8 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵) = (2nd𝑤))
115114eqcomd 2657 . . . . . . 7 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
116112, 115jca 553 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
117 reseq1 5422 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐴) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴))
118117eqeq2d 2661 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ↔ (1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴)))
119 reseq1 5422 . . . . . . . 8 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (𝑧𝐵) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))
120119eqeq2d 2661 . . . . . . 7 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((2nd𝑤) = (𝑧𝐵) ↔ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵)))
121118, 120anbi12d 747 . . . . . 6 (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ ((1st𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐴) ∧ (2nd𝑤) = (((1st𝑤) ∪ (2nd𝑤)) ↾ 𝐵))))
122116, 121syl5ibrcom 237 . . . . 5 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑧 = ((1st𝑤) ∪ (2nd𝑤)) → ((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵))))
12390, 122impbid 202 . . . 4 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (((1st𝑤) = (𝑧𝐴) ∧ (2nd𝑤) = (𝑧𝐵)) ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
12478, 123bitrd 268 . . 3 ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 𝐽)) → (𝑤 = ⟨(𝑧𝐴), (𝑧𝐵)⟩ ↔ 𝑧 = ((1st𝑤) ∪ (2nd𝑤))))
1258, 62, 76, 124f1ocnv2d 6928 . 2 (𝜑 → (𝐺:(𝑋 × 𝑌)–1-1-onto 𝐽𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩)))
126125simprd 478 1 (𝜑𝐺 = (𝑧 𝐽 ↦ ⟨(𝑧𝐴), (𝑧𝐵)⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ⟨cop 4216  ∪ cuni 4468   ↦ cmpt 4762   × cxp 5141  ◡ccnv 5142   ↾ cres 5145   Fn wfn 5921  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Xcixp 7950  ∏tcpt 16146  Topctop 20746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-ixp 7951  df-en 7998  df-fin 8001  df-fi 8358  df-topgen 16151  df-pt 16152  df-top 20747  df-bases 20798 This theorem is referenced by:  ptunhmeo  21659
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