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Theorem f1od2 6132
Description: Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
Hypotheses
Ref Expression
f1od2.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
f1od2.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)
f1od2.3 ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))
f1od2.4 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
Assertion
Ref Expression
f1od2 (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶   𝑥,𝐷,𝑦,𝑧   𝑥,𝐼,𝑦   𝑥,𝐽,𝑦   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧)   𝐼(𝑧)   𝐽(𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem f1od2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 f1od2.2 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)
21ralrimivva 2514 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝐶𝑊)
3 f1od2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fnmpo 6100 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑊𝐹 Fn (𝐴 × 𝐵))
52, 4syl 14 . 2 (𝜑𝐹 Fn (𝐴 × 𝐵))
6 f1od2.3 . . . . . 6 ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))
7 opelxpi 4571 . . . . . 6 ((𝐼𝑋𝐽𝑌) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
86, 7syl 14 . . . . 5 ((𝜑𝑧𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
98ralrimiva 2505 . . . 4 (𝜑 → ∀𝑧𝐷𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
10 eqid 2139 . . . . 5 (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) = (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩)
1110fnmpt 5249 . . . 4 (∀𝑧𝐷𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌) → (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
129, 11syl 14 . . 3 (𝜑 → (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
13 elxp7 6068 . . . . . . . 8 (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)))
1413anbi1i 453 . . . . . . 7 ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
15 anass 398 . . . . . . . . 9 (((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑎 ∈ (V × V) ∧ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)))
16 f1od2.4 . . . . . . . . . . . . 13 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
1716sbcbidv 2967 . . . . . . . . . . . 12 (𝜑 → ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
1817sbcbidv 2967 . . . . . . . . . . 11 (𝜑 → ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
19 sbcan 2951 . . . . . . . . . . . . . 14 ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ∧ [(2nd𝑎) / 𝑦]𝑧 = 𝐶))
20 sbcan 2951 . . . . . . . . . . . . . . . 16 ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ↔ ([(2nd𝑎) / 𝑦]𝑥𝐴[(2nd𝑎) / 𝑦]𝑦𝐵))
21 vex 2689 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
22 2ndexg 6066 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ V → (2nd𝑎) ∈ V)
2321, 22ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2nd𝑎) ∈ V
24 sbcg 2978 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑥𝐴𝑥𝐴))
2523, 24ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑥𝐴𝑥𝐴)
26 sbcel1v 2971 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑦𝐵 ↔ (2nd𝑎) ∈ 𝐵)
2725, 26anbi12i 455 . . . . . . . . . . . . . . . 16 (([(2nd𝑎) / 𝑦]𝑥𝐴[(2nd𝑎) / 𝑦]𝑦𝐵) ↔ (𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵))
2820, 27bitri 183 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵))
29 sbceq2g 3024 . . . . . . . . . . . . . . . 16 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑧 = 𝐶𝑧 = (2nd𝑎) / 𝑦𝐶))
3023, 29ax-mp 5 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦]𝑧 = 𝐶𝑧 = (2nd𝑎) / 𝑦𝐶)
3128, 30anbi12i 455 . . . . . . . . . . . . . 14 (([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ∧ [(2nd𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
3219, 31bitri 183 . . . . . . . . . . . . 13 ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
3332sbcbii 2968 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st𝑎) / 𝑥]((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
34 sbcan 2951 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥]((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶) ↔ ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ [(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶))
35 sbcan 2951 . . . . . . . . . . . . . 14 ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ↔ ([(1st𝑎) / 𝑥]𝑥𝐴[(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵))
36 sbcel1v 2971 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥]𝑥𝐴 ↔ (1st𝑎) ∈ 𝐴)
37 1stexg 6065 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ V → (1st𝑎) ∈ V)
3821, 37ax-mp 5 . . . . . . . . . . . . . . . 16 (1st𝑎) ∈ V
39 sbcg 2978 . . . . . . . . . . . . . . . 16 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵 ↔ (2nd𝑎) ∈ 𝐵))
4038, 39ax-mp 5 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵 ↔ (2nd𝑎) ∈ 𝐵)
4136, 40anbi12i 455 . . . . . . . . . . . . . 14 (([(1st𝑎) / 𝑥]𝑥𝐴[(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵) ↔ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵))
4235, 41bitri 183 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ↔ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵))
43 sbceq2g 3024 . . . . . . . . . . . . . 14 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
4438, 43ax-mp 5 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
4542, 44anbi12i 455 . . . . . . . . . . . 12 (([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ [(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶) ↔ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
4633, 34, 453bitri 205 . . . . . . . . . . 11 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
47 sbcan 2951 . . . . . . . . . . . . . 14 ([(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ ([(2nd𝑎) / 𝑦]𝑧𝐷[(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽)))
48 sbcg 2978 . . . . . . . . . . . . . . . 16 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑧𝐷𝑧𝐷))
4923, 48ax-mp 5 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦]𝑧𝐷𝑧𝐷)
50 sbcan 2951 . . . . . . . . . . . . . . . 16 ([(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽) ↔ ([(2nd𝑎) / 𝑦]𝑥 = 𝐼[(2nd𝑎) / 𝑦]𝑦 = 𝐽))
51 sbcg 2978 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑥 = 𝐼𝑥 = 𝐼))
5223, 51ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑥 = 𝐼𝑥 = 𝐼)
53 sbceq1g 3022 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑦 = 𝐽(2nd𝑎) / 𝑦𝑦 = 𝐽))
5423, 53ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([(2nd𝑎) / 𝑦]𝑦 = 𝐽(2nd𝑎) / 𝑦𝑦 = 𝐽)
55 csbvarg 3030 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑎) ∈ V → (2nd𝑎) / 𝑦𝑦 = (2nd𝑎))
5623, 55ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (2nd𝑎) / 𝑦𝑦 = (2nd𝑎)
5756eqeq1i 2147 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) / 𝑦𝑦 = 𝐽 ↔ (2nd𝑎) = 𝐽)
5854, 57bitri 183 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd𝑎) = 𝐽)
5952, 58anbi12i 455 . . . . . . . . . . . . . . . 16 (([(2nd𝑎) / 𝑦]𝑥 = 𝐼[(2nd𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽))
6050, 59bitri 183 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽))
6149, 60anbi12i 455 . . . . . . . . . . . . . 14 (([(2nd𝑎) / 𝑦]𝑧𝐷[(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
6247, 61bitri 183 . . . . . . . . . . . . 13 ([(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
6362sbcbii 2968 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ [(1st𝑎) / 𝑥](𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
64 sbcan 2951 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥](𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)) ↔ ([(1st𝑎) / 𝑥]𝑧𝐷[(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
65 sbcg 2978 . . . . . . . . . . . . . 14 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑧𝐷𝑧𝐷))
6638, 65ax-mp 5 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥]𝑧𝐷𝑧𝐷)
67 sbcan 2951 . . . . . . . . . . . . . 14 ([(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽) ↔ ([(1st𝑎) / 𝑥]𝑥 = 𝐼[(1st𝑎) / 𝑥](2nd𝑎) = 𝐽))
68 sbceq1g 3022 . . . . . . . . . . . . . . . . 17 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑥 = 𝐼(1st𝑎) / 𝑥𝑥 = 𝐼))
6938, 68ax-mp 5 . . . . . . . . . . . . . . . 16 ([(1st𝑎) / 𝑥]𝑥 = 𝐼(1st𝑎) / 𝑥𝑥 = 𝐼)
70 csbvarg 3030 . . . . . . . . . . . . . . . . . 18 ((1st𝑎) ∈ V → (1st𝑎) / 𝑥𝑥 = (1st𝑎))
7138, 70ax-mp 5 . . . . . . . . . . . . . . . . 17 (1st𝑎) / 𝑥𝑥 = (1st𝑎)
7271eqeq1i 2147 . . . . . . . . . . . . . . . 16 ((1st𝑎) / 𝑥𝑥 = 𝐼 ↔ (1st𝑎) = 𝐼)
7369, 72bitri 183 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st𝑎) = 𝐼)
74 sbcg 2978 . . . . . . . . . . . . . . . 16 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥](2nd𝑎) = 𝐽 ↔ (2nd𝑎) = 𝐽))
7538, 74ax-mp 5 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥](2nd𝑎) = 𝐽 ↔ (2nd𝑎) = 𝐽)
7673, 75anbi12i 455 . . . . . . . . . . . . . 14 (([(1st𝑎) / 𝑥]𝑥 = 𝐼[(1st𝑎) / 𝑥](2nd𝑎) = 𝐽) ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))
7767, 76bitri 183 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽) ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))
7866, 77anbi12i 455 . . . . . . . . . . . 12 (([(1st𝑎) / 𝑥]𝑧𝐷[(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
7963, 64, 783bitri 205 . . . . . . . . . . 11 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
8018, 46, 793bitr3g 221 . . . . . . . . . 10 (𝜑 → ((((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
8180anbi2d 459 . . . . . . . . 9 (𝜑 → ((𝑎 ∈ (V × V) ∧ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))))
8215, 81syl5bb 191 . . . . . . . 8 (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))))
83 xpss 4647 . . . . . . . . . . . 12 (𝑋 × 𝑌) ⊆ (V × V)
84 simprr 521 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 = ⟨𝐼, 𝐽⟩)
858adantrr 470 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
8684, 85eqeltrd 2216 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (𝑋 × 𝑌))
8783, 86sseldi 3095 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (V × V))
8887ex 114 . . . . . . . . . 10 (𝜑 → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) → 𝑎 ∈ (V × V)))
8988pm4.71rd 391 . . . . . . . . 9 (𝜑 → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩))))
90 eqop 6075 . . . . . . . . . . 11 (𝑎 ∈ (V × V) → (𝑎 = ⟨𝐼, 𝐽⟩ ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
9190anbi2d 459 . . . . . . . . . 10 (𝑎 ∈ (V × V) → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
9291pm5.32i 449 . . . . . . . . 9 ((𝑎 ∈ (V × V) ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
9389, 92syl6rbb 196 . . . . . . . 8 (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9482, 93bitrd 187 . . . . . . 7 (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9514, 94syl5bb 191 . . . . . 6 (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9695opabbidv 3994 . . . . 5 (𝜑 → {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)})
97 df-mpo 5779 . . . . . . . 8 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
983, 97eqtri 2160 . . . . . . 7 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9998cnveqi 4714 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
100 nfv 1508 . . . . . . . 8 𝑥 𝑎 ∈ (𝐴 × 𝐵)
101 nfcsb1v 3035 . . . . . . . . 9 𝑥(1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
102101nfeq2 2293 . . . . . . . 8 𝑥 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
103100, 102nfan 1544 . . . . . . 7 𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
104 nfv 1508 . . . . . . . 8 𝑦 𝑎 ∈ (𝐴 × 𝐵)
105 nfcv 2281 . . . . . . . . . 10 𝑦(1st𝑎)
106 nfcsb1v 3035 . . . . . . . . . 10 𝑦(2nd𝑎) / 𝑦𝐶
107105, 106nfcsb 3037 . . . . . . . . 9 𝑦(1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
108107nfeq2 2293 . . . . . . . 8 𝑦 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
109104, 108nfan 1544 . . . . . . 7 𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
110 eleq1 2202 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
111 opelxp 4569 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
112110, 111syl6bb 195 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
113 csbopeq1a 6086 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ → (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶 = 𝐶)
114113eqeq2d 2151 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶𝑧 = 𝐶))
115112, 114anbi12d 464 . . . . . . 7 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)))
116 xpss 4647 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
117116sseli 3093 . . . . . . . 8 (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V))
118117adantr 274 . . . . . . 7 ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) → 𝑎 ∈ (V × V))
119103, 109, 115, 118cnvoprab 6131 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)}
12099, 119eqtri 2160 . . . . 5 𝐹 = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)}
121 df-mpt 3991 . . . . 5 (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) = {⟨𝑧, 𝑎⟩ ∣ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)}
12296, 120, 1213eqtr4g 2197 . . . 4 (𝜑𝐹 = (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩))
123122fneq1d 5213 . . 3 (𝜑 → (𝐹 Fn 𝐷 ↔ (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷))
12412, 123mpbird 166 . 2 (𝜑𝐹 Fn 𝐷)
125 dff1o4 5375 . 2 (𝐹:(𝐴 × 𝐵)–1-1-onto𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐹 Fn 𝐷))
1265, 124, 125sylanbrc 413 1 (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416  Vcvv 2686  [wsbc 2909  csb 3003  cop 3530  {copab 3988  cmpt 3989   × cxp 4537  ccnv 4538   Fn wfn 5118  1-1-ontowf1o 5122  cfv 5123  {coprab 5775  cmpo 5776  1st c1st 6036  2nd c2nd 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039
This theorem is referenced by:  oddpwdc  11858
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