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Theorem xpf1o 8667
Description: Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
xpf1o.1 (𝜑 → (𝑥𝐴𝑋):𝐴1-1-onto𝐵)
xpf1o.2 (𝜑 → (𝑦𝐶𝑌):𝐶1-1-onto𝐷)
Assertion
Ref Expression
xpf1o (𝜑 → (𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑦,𝑋   𝑥,𝐵   𝑦,𝐷   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)   𝐷(𝑥)   𝑋(𝑥)   𝑌(𝑦)

Proof of Theorem xpf1o
Dummy variables 𝑡 𝑠 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 7707 . . . . . 6 (𝑢 ∈ (𝐴 × 𝐶) → (1st𝑢) ∈ 𝐴)
21adantl 485 . . . . 5 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → (1st𝑢) ∈ 𝐴)
3 xpf1o.1 . . . . . . . 8 (𝜑 → (𝑥𝐴𝑋):𝐴1-1-onto𝐵)
4 eqid 2801 . . . . . . . . 9 (𝑥𝐴𝑋) = (𝑥𝐴𝑋)
54f1ompt 6856 . . . . . . . 8 ((𝑥𝐴𝑋):𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝑋𝐵 ∧ ∀𝑧𝐵 ∃!𝑥𝐴 𝑧 = 𝑋))
63, 5sylib 221 . . . . . . 7 (𝜑 → (∀𝑥𝐴 𝑋𝐵 ∧ ∀𝑧𝐵 ∃!𝑥𝐴 𝑧 = 𝑋))
76simpld 498 . . . . . 6 (𝜑 → ∀𝑥𝐴 𝑋𝐵)
87adantr 484 . . . . 5 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑥𝐴 𝑋𝐵)
9 nfcsb1v 3855 . . . . . . 7 𝑥(1st𝑢) / 𝑥𝑋
109nfel1 2974 . . . . . 6 𝑥(1st𝑢) / 𝑥𝑋𝐵
11 csbeq1a 3845 . . . . . . 7 (𝑥 = (1st𝑢) → 𝑋 = (1st𝑢) / 𝑥𝑋)
1211eleq1d 2877 . . . . . 6 (𝑥 = (1st𝑢) → (𝑋𝐵(1st𝑢) / 𝑥𝑋𝐵))
1310, 12rspc 3562 . . . . 5 ((1st𝑢) ∈ 𝐴 → (∀𝑥𝐴 𝑋𝐵(1st𝑢) / 𝑥𝑋𝐵))
142, 8, 13sylc 65 . . . 4 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → (1st𝑢) / 𝑥𝑋𝐵)
15 xp2nd 7708 . . . . . 6 (𝑢 ∈ (𝐴 × 𝐶) → (2nd𝑢) ∈ 𝐶)
1615adantl 485 . . . . 5 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → (2nd𝑢) ∈ 𝐶)
17 xpf1o.2 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑌):𝐶1-1-onto𝐷)
18 eqid 2801 . . . . . . . . 9 (𝑦𝐶𝑌) = (𝑦𝐶𝑌)
1918f1ompt 6856 . . . . . . . 8 ((𝑦𝐶𝑌):𝐶1-1-onto𝐷 ↔ (∀𝑦𝐶 𝑌𝐷 ∧ ∀𝑤𝐷 ∃!𝑦𝐶 𝑤 = 𝑌))
2017, 19sylib 221 . . . . . . 7 (𝜑 → (∀𝑦𝐶 𝑌𝐷 ∧ ∀𝑤𝐷 ∃!𝑦𝐶 𝑤 = 𝑌))
2120simpld 498 . . . . . 6 (𝜑 → ∀𝑦𝐶 𝑌𝐷)
2221adantr 484 . . . . 5 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → ∀𝑦𝐶 𝑌𝐷)
23 nfcsb1v 3855 . . . . . . 7 𝑦(2nd𝑢) / 𝑦𝑌
2423nfel1 2974 . . . . . 6 𝑦(2nd𝑢) / 𝑦𝑌𝐷
25 csbeq1a 3845 . . . . . . 7 (𝑦 = (2nd𝑢) → 𝑌 = (2nd𝑢) / 𝑦𝑌)
2625eleq1d 2877 . . . . . 6 (𝑦 = (2nd𝑢) → (𝑌𝐷(2nd𝑢) / 𝑦𝑌𝐷))
2724, 26rspc 3562 . . . . 5 ((2nd𝑢) ∈ 𝐶 → (∀𝑦𝐶 𝑌𝐷(2nd𝑢) / 𝑦𝑌𝐷))
2816, 22, 27sylc 65 . . . 4 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → (2nd𝑢) / 𝑦𝑌𝐷)
2914, 28opelxpd 5561 . . 3 ((𝜑𝑢 ∈ (𝐴 × 𝐶)) → ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ∈ (𝐵 × 𝐷))
3029ralrimiva 3152 . 2 (𝜑 → ∀𝑢 ∈ (𝐴 × 𝐶)⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ∈ (𝐵 × 𝐷))
316simprd 499 . . . . . . . . . 10 (𝜑 → ∀𝑧𝐵 ∃!𝑥𝐴 𝑧 = 𝑋)
3231r19.21bi 3176 . . . . . . . . 9 ((𝜑𝑧𝐵) → ∃!𝑥𝐴 𝑧 = 𝑋)
33 reu6 3668 . . . . . . . . 9 (∃!𝑥𝐴 𝑧 = 𝑋 ↔ ∃𝑠𝐴𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠))
3432, 33sylib 221 . . . . . . . 8 ((𝜑𝑧𝐵) → ∃𝑠𝐴𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠))
3520simprd 499 . . . . . . . . . 10 (𝜑 → ∀𝑤𝐷 ∃!𝑦𝐶 𝑤 = 𝑌)
3635r19.21bi 3176 . . . . . . . . 9 ((𝜑𝑤𝐷) → ∃!𝑦𝐶 𝑤 = 𝑌)
37 reu6 3668 . . . . . . . . 9 (∃!𝑦𝐶 𝑤 = 𝑌 ↔ ∃𝑡𝐶𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡))
3836, 37sylib 221 . . . . . . . 8 ((𝜑𝑤𝐷) → ∃𝑡𝐶𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡))
3934, 38anim12dan 621 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐷)) → (∃𝑠𝐴𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∃𝑡𝐶𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)))
40 reeanv 3323 . . . . . . . 8 (∃𝑠𝐴𝑡𝐶 (∀𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)) ↔ (∃𝑠𝐴𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∃𝑡𝐶𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)))
41 pm4.38 637 . . . . . . . . . . . . . . 15 (((𝑧 = 𝑋𝑥 = 𝑠) ∧ (𝑤 = 𝑌𝑦 = 𝑡)) → ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
4241ex 416 . . . . . . . . . . . . . 14 ((𝑧 = 𝑋𝑥 = 𝑠) → ((𝑤 = 𝑌𝑦 = 𝑡) → ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
4342ralimdv 3148 . . . . . . . . . . . . 13 ((𝑧 = 𝑋𝑥 = 𝑠) → (∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡) → ∀𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
4443com12 32 . . . . . . . . . . . 12 (∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡) → ((𝑧 = 𝑋𝑥 = 𝑠) → ∀𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
4544ralimdv 3148 . . . . . . . . . . 11 (∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡) → (∀𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) → ∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
4645impcom 411 . . . . . . . . . 10 ((∀𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)) → ∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
4746reximi 3209 . . . . . . . . 9 (∃𝑡𝐶 (∀𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)) → ∃𝑡𝐶𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
4847reximi 3209 . . . . . . . 8 (∃𝑠𝐴𝑡𝐶 (∀𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∀𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)) → ∃𝑠𝐴𝑡𝐶𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
4940, 48sylbir 238 . . . . . . 7 ((∃𝑠𝐴𝑥𝐴 (𝑧 = 𝑋𝑥 = 𝑠) ∧ ∃𝑡𝐶𝑦𝐶 (𝑤 = 𝑌𝑦 = 𝑡)) → ∃𝑠𝐴𝑡𝐶𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
5039, 49syl 17 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐷)) → ∃𝑠𝐴𝑡𝐶𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
51 vex 3447 . . . . . . . . . . . . . . 15 𝑠 ∈ V
52 vex 3447 . . . . . . . . . . . . . . 15 𝑡 ∈ V
5351, 52op1std 7685 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑠, 𝑡⟩ → (1st𝑢) = 𝑠)
5453csbeq1d 3835 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑠, 𝑡⟩ → (1st𝑢) / 𝑥𝑋 = 𝑠 / 𝑥𝑋)
5554eqeq2d 2812 . . . . . . . . . . . 12 (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑧 = (1st𝑢) / 𝑥𝑋𝑧 = 𝑠 / 𝑥𝑋))
5651, 52op2ndd 7686 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑠, 𝑡⟩ → (2nd𝑢) = 𝑡)
5756csbeq1d 3835 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑠, 𝑡⟩ → (2nd𝑢) / 𝑦𝑌 = 𝑡 / 𝑦𝑌)
5857eqeq2d 2812 . . . . . . . . . . . 12 (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑤 = (2nd𝑢) / 𝑦𝑌𝑤 = 𝑡 / 𝑦𝑌))
5955, 58anbi12d 633 . . . . . . . . . . 11 (𝑢 = ⟨𝑠, 𝑡⟩ → ((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ (𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌)))
60 eqeq1 2805 . . . . . . . . . . 11 (𝑢 = ⟨𝑠, 𝑡⟩ → (𝑢 = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
6159, 60bibi12d 349 . . . . . . . . . 10 (𝑢 = ⟨𝑠, 𝑡⟩ → (((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣) ↔ ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
6261ralxp 5680 . . . . . . . . 9 (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑠𝐴𝑡𝐶 ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
63 nfv 1915 . . . . . . . . . 10 𝑠𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
64 nfcv 2958 . . . . . . . . . . 11 𝑥𝐶
65 nfcsb1v 3855 . . . . . . . . . . . . . 14 𝑥𝑠 / 𝑥𝑋
6665nfeq2 2975 . . . . . . . . . . . . 13 𝑥 𝑧 = 𝑠 / 𝑥𝑋
67 nfv 1915 . . . . . . . . . . . . 13 𝑥 𝑤 = 𝑡 / 𝑦𝑌
6866, 67nfan 1900 . . . . . . . . . . . 12 𝑥(𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌)
69 nfv 1915 . . . . . . . . . . . 12 𝑥𝑠, 𝑡⟩ = 𝑣
7068, 69nfbi 1904 . . . . . . . . . . 11 𝑥((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
7164, 70nfralw 3192 . . . . . . . . . 10 𝑥𝑡𝐶 ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)
72 nfv 1915 . . . . . . . . . . . 12 𝑡((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣)
73 nfv 1915 . . . . . . . . . . . . . 14 𝑦 𝑧 = 𝑋
74 nfcsb1v 3855 . . . . . . . . . . . . . . 15 𝑦𝑡 / 𝑦𝑌
7574nfeq2 2975 . . . . . . . . . . . . . 14 𝑦 𝑤 = 𝑡 / 𝑦𝑌
7673, 75nfan 1900 . . . . . . . . . . . . 13 𝑦(𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌)
77 nfv 1915 . . . . . . . . . . . . 13 𝑦𝑥, 𝑡⟩ = 𝑣
7876, 77nfbi 1904 . . . . . . . . . . . 12 𝑦((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)
79 csbeq1a 3845 . . . . . . . . . . . . . . 15 (𝑦 = 𝑡𝑌 = 𝑡 / 𝑦𝑌)
8079eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → (𝑤 = 𝑌𝑤 = 𝑡 / 𝑦𝑌))
8180anbi2d 631 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌)))
82 opeq2 4768 . . . . . . . . . . . . . 14 (𝑦 = 𝑡 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑡⟩)
8382eqeq1d 2803 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
8481, 83bibi12d 349 . . . . . . . . . . . 12 (𝑦 = 𝑡 → (((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣)))
8572, 78, 84cbvralw 3390 . . . . . . . . . . 11 (∀𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡𝐶 ((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣))
86 csbeq1a 3845 . . . . . . . . . . . . . . 15 (𝑥 = 𝑠𝑋 = 𝑠 / 𝑥𝑋)
8786eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑥 = 𝑠 → (𝑧 = 𝑋𝑧 = 𝑠 / 𝑥𝑋))
8887anbi1d 632 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → ((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ (𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌)))
89 opeq1 4766 . . . . . . . . . . . . . 14 (𝑥 = 𝑠 → ⟨𝑥, 𝑡⟩ = ⟨𝑠, 𝑡⟩)
9089eqeq1d 2803 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → (⟨𝑥, 𝑡⟩ = 𝑣 ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
9188, 90bibi12d 349 . . . . . . . . . . . 12 (𝑥 = 𝑠 → (((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣) ↔ ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
9291ralbidv 3165 . . . . . . . . . . 11 (𝑥 = 𝑠 → (∀𝑡𝐶 ((𝑧 = 𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑥, 𝑡⟩ = 𝑣) ↔ ∀𝑡𝐶 ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
9385, 92syl5bb 286 . . . . . . . . . 10 (𝑥 = 𝑠 → (∀𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑡𝐶 ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣)))
9463, 71, 93cbvralw 3390 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑠𝐴𝑡𝐶 ((𝑧 = 𝑠 / 𝑥𝑋𝑤 = 𝑡 / 𝑦𝑌) ↔ ⟨𝑠, 𝑡⟩ = 𝑣))
9562, 94bitr4i 281 . . . . . . . 8 (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣))
96 eqeq2 2813 . . . . . . . . . . 11 (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩))
97 vex 3447 . . . . . . . . . . . 12 𝑥 ∈ V
98 vex 3447 . . . . . . . . . . . 12 𝑦 ∈ V
9997, 98opth 5336 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ = ⟨𝑠, 𝑡⟩ ↔ (𝑥 = 𝑠𝑦 = 𝑡))
10096, 99syl6bb 290 . . . . . . . . . 10 (𝑣 = ⟨𝑠, 𝑡⟩ → (⟨𝑥, 𝑦⟩ = 𝑣 ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
101100bibi2d 346 . . . . . . . . 9 (𝑣 = ⟨𝑠, 𝑡⟩ → (((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
1021012ralbidv 3167 . . . . . . . 8 (𝑣 = ⟨𝑠, 𝑡⟩ → (∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ ⟨𝑥, 𝑦⟩ = 𝑣) ↔ ∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
10395, 102syl5bb 286 . . . . . . 7 (𝑣 = ⟨𝑠, 𝑡⟩ → (∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣) ↔ ∀𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡))))
104103rexxp 5681 . . . . . 6 (∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣) ↔ ∃𝑠𝐴𝑡𝐶𝑥𝐴𝑦𝐶 ((𝑧 = 𝑋𝑤 = 𝑌) ↔ (𝑥 = 𝑠𝑦 = 𝑡)))
10550, 104sylibr 237 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐷)) → ∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣))
106 reu6 3668 . . . . 5 (∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ ∃𝑣 ∈ (𝐴 × 𝐶)∀𝑢 ∈ (𝐴 × 𝐶)((𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌) ↔ 𝑢 = 𝑣))
107105, 106sylibr 237 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐷)) → ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌))
108107ralrimivva 3159 . . 3 (𝜑 → ∀𝑧𝐵𝑤𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌))
109 eqeq1 2805 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩))
110 vex 3447 . . . . . . 7 𝑧 ∈ V
111 vex 3447 . . . . . . 7 𝑤 ∈ V
112110, 111opth 5336 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ↔ (𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌))
113109, 112syl6bb 290 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ↔ (𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌)))
114113reubidv 3345 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → (∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ↔ ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌)))
115114ralxp 5680 . . 3 (∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ↔ ∀𝑧𝐵𝑤𝐷 ∃!𝑢 ∈ (𝐴 × 𝐶)(𝑧 = (1st𝑢) / 𝑥𝑋𝑤 = (2nd𝑢) / 𝑦𝑌))
116108, 115sylibr 237 . 2 (𝜑 → ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩)
117 nfcv 2958 . . . . 5 𝑧𝑋, 𝑌
118 nfcv 2958 . . . . 5 𝑤𝑋, 𝑌
119 nfcsb1v 3855 . . . . . 6 𝑥𝑧 / 𝑥𝑋
120 nfcv 2958 . . . . . 6 𝑥𝑤 / 𝑦𝑌
121119, 120nfop 4784 . . . . 5 𝑥𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌
122 nfcv 2958 . . . . . 6 𝑦𝑧 / 𝑥𝑋
123 nfcsb1v 3855 . . . . . 6 𝑦𝑤 / 𝑦𝑌
124122, 123nfop 4784 . . . . 5 𝑦𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌
125 csbeq1a 3845 . . . . . 6 (𝑥 = 𝑧𝑋 = 𝑧 / 𝑥𝑋)
126 csbeq1a 3845 . . . . . 6 (𝑦 = 𝑤𝑌 = 𝑤 / 𝑦𝑌)
127 opeq12 4770 . . . . . 6 ((𝑋 = 𝑧 / 𝑥𝑋𝑌 = 𝑤 / 𝑦𝑌) → ⟨𝑋, 𝑌⟩ = ⟨𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌⟩)
128125, 126, 127syl2an 598 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → ⟨𝑋, 𝑌⟩ = ⟨𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌⟩)
129117, 118, 121, 124, 128cbvmpo 7231 . . . 4 (𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑧𝐴, 𝑤𝐶 ↦ ⟨𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌⟩)
130110, 111op1std 7685 . . . . . . 7 (𝑢 = ⟨𝑧, 𝑤⟩ → (1st𝑢) = 𝑧)
131130csbeq1d 3835 . . . . . 6 (𝑢 = ⟨𝑧, 𝑤⟩ → (1st𝑢) / 𝑥𝑋 = 𝑧 / 𝑥𝑋)
132110, 111op2ndd 7686 . . . . . . 7 (𝑢 = ⟨𝑧, 𝑤⟩ → (2nd𝑢) = 𝑤)
133132csbeq1d 3835 . . . . . 6 (𝑢 = ⟨𝑧, 𝑤⟩ → (2nd𝑢) / 𝑦𝑌 = 𝑤 / 𝑦𝑌)
134131, 133opeq12d 4776 . . . . 5 (𝑢 = ⟨𝑧, 𝑤⟩ → ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ = ⟨𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌⟩)
135134mpompt 7249 . . . 4 (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩) = (𝑧𝐴, 𝑤𝐶 ↦ ⟨𝑧 / 𝑥𝑋, 𝑤 / 𝑦𝑌⟩)
136129, 135eqtr4i 2827 . . 3 (𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩) = (𝑢 ∈ (𝐴 × 𝐶) ↦ ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩)
137136f1ompt 6856 . 2 ((𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷) ↔ (∀𝑢 ∈ (𝐴 × 𝐶)⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩ ∈ (𝐵 × 𝐷) ∧ ∀𝑣 ∈ (𝐵 × 𝐷)∃!𝑢 ∈ (𝐴 × 𝐶)𝑣 = ⟨(1st𝑢) / 𝑥𝑋, (2nd𝑢) / 𝑦𝑌⟩))
13830, 116, 137sylanbrc 586 1 (𝜑 → (𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  csb 3831  cop 4534  cmpt 5113   × cxp 5521  1-1-ontowf1o 6327  cfv 6328  cmpo 7141  1st c1st 7673  2nd c2nd 7674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676
This theorem is referenced by:  infxpenc  9433  pwfseqlem5  10078
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