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Theorem ptuncnv 23311
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 = βˆͺ 𝐾
ptunhmeo.y π‘Œ = βˆͺ 𝐿
ptunhmeo.j 𝐽 = (∏tβ€˜πΉ)
ptunhmeo.k 𝐾 = (∏tβ€˜(𝐹 β†Ύ 𝐴))
ptunhmeo.l 𝐿 = (∏tβ€˜(𝐹 β†Ύ 𝐡))
ptunhmeo.g 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (π‘₯ βˆͺ 𝑦))
ptunhmeo.c (πœ‘ β†’ 𝐢 ∈ 𝑉)
ptunhmeo.f (πœ‘ β†’ 𝐹:𝐢⟢Top)
ptunhmeo.u (πœ‘ β†’ 𝐢 = (𝐴 βˆͺ 𝐡))
ptunhmeo.i (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)
Assertion
Ref Expression
ptuncnv (πœ‘ β†’ ◑𝐺 = (𝑧 ∈ βˆͺ 𝐽 ↦ ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐴   π‘₯,𝐡,𝑦,𝑧   𝑧,𝐺   πœ‘,π‘₯,𝑦,𝑧   π‘₯,𝐢,𝑦,𝑧   π‘₯,𝐹,𝑦,𝑧   π‘₯,𝐽,𝑦,𝑧   π‘₯,𝐾,𝑦,𝑧   π‘₯,𝐿,𝑦,𝑧   𝑧,𝑉   π‘₯,𝑋,𝑦,𝑧   π‘₯,π‘Œ,𝑦,𝑧
Allowed substitution hints:   𝐺(π‘₯,𝑦)   𝑉(π‘₯,𝑦)
Proof of Theorem ptuncnv
Dummy variables π‘˜ 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptunhmeo.g . . . 4 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (π‘₯ βˆͺ 𝑦))
2 vex 3479 . . . . . . 7 π‘₯ ∈ V
3 vex 3479 . . . . . . 7 𝑦 ∈ V
42, 3op1std 7985 . . . . . 6 (𝑀 = ⟨π‘₯, π‘¦βŸ© β†’ (1st β€˜π‘€) = π‘₯)
52, 3op2ndd 7986 . . . . . 6 (𝑀 = ⟨π‘₯, π‘¦βŸ© β†’ (2nd β€˜π‘€) = 𝑦)
64, 5uneq12d 4165 . . . . 5 (𝑀 = ⟨π‘₯, π‘¦βŸ© β†’ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) = (π‘₯ βˆͺ 𝑦))
76mpompt 7522 . . . 4 (𝑀 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€))) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (π‘₯ βˆͺ 𝑦))
81, 7eqtr4i 2764 . . 3 𝐺 = (𝑀 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)))
9 xp1st 8007 . . . . . . 7 (𝑀 ∈ (𝑋 Γ— π‘Œ) β†’ (1st β€˜π‘€) ∈ 𝑋)
109adantl 483 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (1st β€˜π‘€) ∈ 𝑋)
11 ixpeq2 8905 . . . . . . . . . 10 (βˆ€π‘˜ ∈ 𝐴 βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = βˆͺ (πΉβ€˜π‘˜) β†’ Xπ‘˜ ∈ 𝐴 βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜))
12 fvres 6911 . . . . . . . . . . 11 (π‘˜ ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = (πΉβ€˜π‘˜))
1312unieqd 4923 . . . . . . . . . 10 (π‘˜ ∈ 𝐴 β†’ βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = βˆͺ (πΉβ€˜π‘˜))
1411, 13mprg 3068 . . . . . . . . 9 Xπ‘˜ ∈ 𝐴 βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜)
15 ptunhmeo.c . . . . . . . . . . 11 (πœ‘ β†’ 𝐢 ∈ 𝑉)
16 ssun1 4173 . . . . . . . . . . . 12 𝐴 βŠ† (𝐴 βˆͺ 𝐡)
17 ptunhmeo.u . . . . . . . . . . . 12 (πœ‘ β†’ 𝐢 = (𝐴 βˆͺ 𝐡))
1816, 17sseqtrrid 4036 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 βŠ† 𝐢)
1915, 18ssexd 5325 . . . . . . . . . 10 (πœ‘ β†’ 𝐴 ∈ V)
20 ptunhmeo.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:𝐢⟢Top)
2120, 18fssresd 6759 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 β†Ύ 𝐴):𝐴⟢Top)
22 ptunhmeo.k . . . . . . . . . . 11 𝐾 = (∏tβ€˜(𝐹 β†Ύ 𝐴))
2322ptuni 23098 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝐹 β†Ύ 𝐴):𝐴⟢Top) β†’ Xπ‘˜ ∈ 𝐴 βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = βˆͺ 𝐾)
2419, 21, 23syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ Xπ‘˜ ∈ 𝐴 βˆͺ ((𝐹 β†Ύ 𝐴)β€˜π‘˜) = βˆͺ 𝐾)
2514, 24eqtr3id 2787 . . . . . . . 8 (πœ‘ β†’ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) = βˆͺ 𝐾)
26 ptunhmeo.x . . . . . . . 8 𝑋 = βˆͺ 𝐾
2725, 26eqtr4di 2791 . . . . . . 7 (πœ‘ β†’ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) = 𝑋)
2827adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) = 𝑋)
2910, 28eleqtrrd 2837 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (1st β€˜π‘€) ∈ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜))
30 xp2nd 8008 . . . . . . 7 (𝑀 ∈ (𝑋 Γ— π‘Œ) β†’ (2nd β€˜π‘€) ∈ π‘Œ)
3130adantl 483 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (2nd β€˜π‘€) ∈ π‘Œ)
3217eqcomd 2739 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 βˆͺ 𝐡) = 𝐢)
33 ptunhmeo.i . . . . . . . . . . 11 (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)
34 uneqdifeq 4493 . . . . . . . . . . 11 ((𝐴 βŠ† 𝐢 ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((𝐴 βˆͺ 𝐡) = 𝐢 ↔ (𝐢 βˆ– 𝐴) = 𝐡))
3518, 33, 34syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ ((𝐴 βˆͺ 𝐡) = 𝐢 ↔ (𝐢 βˆ– 𝐴) = 𝐡))
3632, 35mpbid 231 . . . . . . . . 9 (πœ‘ β†’ (𝐢 βˆ– 𝐴) = 𝐡)
3736ixpeq1d 8903 . . . . . . . 8 (πœ‘ β†’ Xπ‘˜ ∈ (𝐢 βˆ– 𝐴)βˆͺ (πΉβ€˜π‘˜) = Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜))
38 ixpeq2 8905 . . . . . . . . . . 11 (βˆ€π‘˜ ∈ 𝐡 βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = βˆͺ (πΉβ€˜π‘˜) β†’ Xπ‘˜ ∈ 𝐡 βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜))
39 fvres 6911 . . . . . . . . . . . 12 (π‘˜ ∈ 𝐡 β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = (πΉβ€˜π‘˜))
4039unieqd 4923 . . . . . . . . . . 11 (π‘˜ ∈ 𝐡 β†’ βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = βˆͺ (πΉβ€˜π‘˜))
4138, 40mprg 3068 . . . . . . . . . 10 Xπ‘˜ ∈ 𝐡 βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜)
42 ssun2 4174 . . . . . . . . . . . . 13 𝐡 βŠ† (𝐴 βˆͺ 𝐡)
4342, 17sseqtrrid 4036 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐡 βŠ† 𝐢)
4415, 43ssexd 5325 . . . . . . . . . . 11 (πœ‘ β†’ 𝐡 ∈ V)
4520, 43fssresd 6759 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 β†Ύ 𝐡):𝐡⟢Top)
46 ptunhmeo.l . . . . . . . . . . . 12 𝐿 = (∏tβ€˜(𝐹 β†Ύ 𝐡))
4746ptuni 23098 . . . . . . . . . . 11 ((𝐡 ∈ V ∧ (𝐹 β†Ύ 𝐡):𝐡⟢Top) β†’ Xπ‘˜ ∈ 𝐡 βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = βˆͺ 𝐿)
4844, 45, 47syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ Xπ‘˜ ∈ 𝐡 βˆͺ ((𝐹 β†Ύ 𝐡)β€˜π‘˜) = βˆͺ 𝐿)
4941, 48eqtr3id 2787 . . . . . . . . 9 (πœ‘ β†’ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜) = βˆͺ 𝐿)
50 ptunhmeo.y . . . . . . . . 9 π‘Œ = βˆͺ 𝐿
5149, 50eqtr4di 2791 . . . . . . . 8 (πœ‘ β†’ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜) = π‘Œ)
5237, 51eqtrd 2773 . . . . . . 7 (πœ‘ β†’ Xπ‘˜ ∈ (𝐢 βˆ– 𝐴)βˆͺ (πΉβ€˜π‘˜) = π‘Œ)
5352adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ Xπ‘˜ ∈ (𝐢 βˆ– 𝐴)βˆͺ (πΉβ€˜π‘˜) = π‘Œ)
5431, 53eleqtrrd 2837 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (2nd β€˜π‘€) ∈ Xπ‘˜ ∈ (𝐢 βˆ– 𝐴)βˆͺ (πΉβ€˜π‘˜))
5518adantr 482 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ 𝐴 βŠ† 𝐢)
56 undifixp 8928 . . . . 5 (((1st β€˜π‘€) ∈ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) ∧ (2nd β€˜π‘€) ∈ Xπ‘˜ ∈ (𝐢 βˆ– 𝐴)βˆͺ (πΉβ€˜π‘˜) ∧ 𝐴 βŠ† 𝐢) β†’ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜))
5729, 54, 55, 56syl3anc 1372 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜))
58 ptunhmeo.j . . . . . . 7 𝐽 = (∏tβ€˜πΉ)
5958ptuni 23098 . . . . . 6 ((𝐢 ∈ 𝑉 ∧ 𝐹:𝐢⟢Top) β†’ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜) = βˆͺ 𝐽)
6015, 20, 59syl2anc 585 . . . . 5 (πœ‘ β†’ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜) = βˆͺ 𝐽)
6160adantr 482 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜) = βˆͺ 𝐽)
6257, 61eleqtrd 2836 . . 3 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) ∈ βˆͺ 𝐽)
6318adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ 𝐴 βŠ† 𝐢)
6460eleq2d 2820 . . . . . . 7 (πœ‘ β†’ (𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜) ↔ 𝑧 ∈ βˆͺ 𝐽))
6564biimpar 479 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ 𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜))
66 resixp 8927 . . . . . 6 ((𝐴 βŠ† 𝐢 ∧ 𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜)) β†’ (𝑧 β†Ύ 𝐴) ∈ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜))
6763, 65, 66syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ (𝑧 β†Ύ 𝐴) ∈ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜))
6827adantr 482 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) = 𝑋)
6967, 68eleqtrd 2836 . . . 4 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ (𝑧 β†Ύ 𝐴) ∈ 𝑋)
7043adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ 𝐡 βŠ† 𝐢)
71 resixp 8927 . . . . . 6 ((𝐡 βŠ† 𝐢 ∧ 𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜)) β†’ (𝑧 β†Ύ 𝐡) ∈ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜))
7270, 65, 71syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ (𝑧 β†Ύ 𝐡) ∈ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜))
7351adantr 482 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜) = π‘Œ)
7472, 73eleqtrd 2836 . . . 4 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ (𝑧 β†Ύ 𝐡) ∈ π‘Œ)
7569, 74opelxpd 5716 . . 3 ((πœ‘ ∧ 𝑧 ∈ βˆͺ 𝐽) β†’ ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩ ∈ (𝑋 Γ— π‘Œ))
76 eqop 8017 . . . . 5 (𝑀 ∈ (𝑋 Γ— π‘Œ) β†’ (𝑀 = ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩ ↔ ((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡))))
7776ad2antrl 727 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝑀 = ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩ ↔ ((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡))))
7865adantrl 715 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜))
79 ixpfn 8897 . . . . . . . . 9 (𝑧 ∈ Xπ‘˜ ∈ 𝐢 βˆͺ (πΉβ€˜π‘˜) β†’ 𝑧 Fn 𝐢)
80 fnresdm 6670 . . . . . . . . 9 (𝑧 Fn 𝐢 β†’ (𝑧 β†Ύ 𝐢) = 𝑧)
8178, 79, 803syl 18 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝑧 β†Ύ 𝐢) = 𝑧)
8217reseq2d 5982 . . . . . . . . 9 (πœ‘ β†’ (𝑧 β†Ύ 𝐢) = (𝑧 β†Ύ (𝐴 βˆͺ 𝐡)))
8382adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝑧 β†Ύ 𝐢) = (𝑧 β†Ύ (𝐴 βˆͺ 𝐡)))
8481, 83eqtr3d 2775 . . . . . . 7 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝑧 = (𝑧 β†Ύ (𝐴 βˆͺ 𝐡)))
85 resundi 5996 . . . . . . 7 (𝑧 β†Ύ (𝐴 βˆͺ 𝐡)) = ((𝑧 β†Ύ 𝐴) βˆͺ (𝑧 β†Ύ 𝐡))
8684, 85eqtrdi 2789 . . . . . 6 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ 𝑧 = ((𝑧 β†Ύ 𝐴) βˆͺ (𝑧 β†Ύ 𝐡)))
87 uneq12 4159 . . . . . . 7 (((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡)) β†’ ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) = ((𝑧 β†Ύ 𝐴) βˆͺ (𝑧 β†Ύ 𝐡)))
8887eqeq2d 2744 . . . . . 6 (((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡)) β†’ (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) ↔ 𝑧 = ((𝑧 β†Ύ 𝐴) βˆͺ (𝑧 β†Ύ 𝐡))))
8986, 88syl5ibrcom 246 . . . . 5 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡)) β†’ 𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€))))
90 ixpfn 8897 . . . . . . . . . . . 12 ((1st β€˜π‘€) ∈ Xπ‘˜ ∈ 𝐴 βˆͺ (πΉβ€˜π‘˜) β†’ (1st β€˜π‘€) Fn 𝐴)
9129, 90syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (1st β€˜π‘€) Fn 𝐴)
9291adantrr 716 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (1st β€˜π‘€) Fn 𝐴)
93 dffn2 6720 . . . . . . . . . 10 ((1st β€˜π‘€) Fn 𝐴 ↔ (1st β€˜π‘€):𝐴⟢V)
9492, 93sylib 217 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (1st β€˜π‘€):𝐴⟢V)
9551adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜) = π‘Œ)
9631, 95eleqtrrd 2837 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (2nd β€˜π‘€) ∈ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜))
97 ixpfn 8897 . . . . . . . . . . . 12 ((2nd β€˜π‘€) ∈ Xπ‘˜ ∈ 𝐡 βˆͺ (πΉβ€˜π‘˜) β†’ (2nd β€˜π‘€) Fn 𝐡)
9896, 97syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ (𝑋 Γ— π‘Œ)) β†’ (2nd β€˜π‘€) Fn 𝐡)
9998adantrr 716 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (2nd β€˜π‘€) Fn 𝐡)
100 dffn2 6720 . . . . . . . . . 10 ((2nd β€˜π‘€) Fn 𝐡 ↔ (2nd β€˜π‘€):𝐡⟢V)
10199, 100sylib 217 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (2nd β€˜π‘€):𝐡⟢V)
102 res0 5986 . . . . . . . . . . 11 ((1st β€˜π‘€) β†Ύ βˆ…) = βˆ…
103 res0 5986 . . . . . . . . . . 11 ((2nd β€˜π‘€) β†Ύ βˆ…) = βˆ…
104102, 103eqtr4i 2764 . . . . . . . . . 10 ((1st β€˜π‘€) β†Ύ βˆ…) = ((2nd β€˜π‘€) β†Ύ βˆ…)
10533adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝐴 ∩ 𝐡) = βˆ…)
106105reseq2d 5982 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ ((1st β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)) = ((1st β€˜π‘€) β†Ύ βˆ…))
107105reseq2d 5982 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ ((2nd β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)) = ((2nd β€˜π‘€) β†Ύ βˆ…))
108104, 106, 1073eqtr4a 2799 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ ((1st β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)) = ((2nd β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)))
109 fresaunres1 6765 . . . . . . . . 9 (((1st β€˜π‘€):𝐴⟢V ∧ (2nd β€˜π‘€):𝐡⟢V ∧ ((1st β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)) = ((2nd β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡))) β†’ (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴) = (1st β€˜π‘€))
11094, 101, 108, 109syl3anc 1372 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴) = (1st β€˜π‘€))
111110eqcomd 2739 . . . . . . 7 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (1st β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴))
112 fresaunres2 6764 . . . . . . . . 9 (((1st β€˜π‘€):𝐴⟢V ∧ (2nd β€˜π‘€):𝐡⟢V ∧ ((1st β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡)) = ((2nd β€˜π‘€) β†Ύ (𝐴 ∩ 𝐡))) β†’ (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡) = (2nd β€˜π‘€))
11394, 101, 108, 112syl3anc 1372 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡) = (2nd β€˜π‘€))
114113eqcomd 2739 . . . . . . 7 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (2nd β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡))
115111, 114jca 513 . . . . . 6 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ ((1st β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡)))
116 reseq1 5976 . . . . . . . 8 (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ (𝑧 β†Ύ 𝐴) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴))
117116eqeq2d 2744 . . . . . . 7 (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ ((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ↔ (1st β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴)))
118 reseq1 5976 . . . . . . . 8 (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ (𝑧 β†Ύ 𝐡) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡))
119118eqeq2d 2744 . . . . . . 7 (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ ((2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡) ↔ (2nd β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡)))
120117, 119anbi12d 632 . . . . . 6 (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ (((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡)) ↔ ((1st β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†Ύ 𝐡))))
121115, 120syl5ibrcom 246 . . . . 5 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€)) β†’ ((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡))))
12289, 121impbid 211 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (((1st β€˜π‘€) = (𝑧 β†Ύ 𝐴) ∧ (2nd β€˜π‘€) = (𝑧 β†Ύ 𝐡)) ↔ 𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€))))
12377, 122bitrd 279 . . 3 ((πœ‘ ∧ (𝑀 ∈ (𝑋 Γ— π‘Œ) ∧ 𝑧 ∈ βˆͺ 𝐽)) β†’ (𝑀 = ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩ ↔ 𝑧 = ((1st β€˜π‘€) βˆͺ (2nd β€˜π‘€))))
1248, 62, 75, 123f1ocnv2d 7659 . 2 (πœ‘ β†’ (𝐺:(𝑋 Γ— π‘Œ)–1-1-ontoβ†’βˆͺ 𝐽 ∧ ◑𝐺 = (𝑧 ∈ βˆͺ 𝐽 ↦ ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩)))
125124simprd 497 1 (πœ‘ β†’ ◑𝐺 = (𝑧 ∈ βˆͺ 𝐽 ↦ ⟨(𝑧 β†Ύ 𝐴), (𝑧 β†Ύ 𝐡)⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  βŸ¨cop 4635  βˆͺ cuni 4909   ↦ cmpt 5232   Γ— cxp 5675  β—‘ccnv 5676   β†Ύ cres 5679   Fn wfn 6539  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  Xcixp 8891  βˆtcpt 17384  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-ixp 8892  df-en 8940  df-fin 8943  df-fi 9406  df-topgen 17389  df-pt 17390  df-top 22396  df-bases 22449
This theorem is referenced by:  ptunhmeo  23312
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