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Theorem f1od2 5982
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 2455 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝐶𝑊)
3 f1od2.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43fnmpt2 5954 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑊𝐹 Fn (𝐴 × 𝐵))
52, 4syl 14 . 2 (𝜑𝐹 Fn (𝐴 × 𝐵))
6 f1od2.3 . . . . . 6 ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))
7 opelxpi 4459 . . . . . 6 ((𝐼𝑋𝐽𝑌) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
86, 7syl 14 . . . . 5 ((𝜑𝑧𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
98ralrimiva 2446 . . . 4 (𝜑 → ∀𝑧𝐷𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
10 eqid 2088 . . . . 5 (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) = (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩)
1110fnmpt 5126 . . . 4 (∀𝑧𝐷𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌) → (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
129, 11syl 14 . . 3 (𝜑 → (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
13 elxp7 5923 . . . . . . . 8 (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)))
1413anbi1i 446 . . . . . . 7 ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
15 anass 393 . . . . . . . . 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 2895 . . . . . . . . . . . 12 (𝜑 → ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
1817sbcbidv 2895 . . . . . . . . . . 11 (𝜑 → ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))
19 sbcan 2879 . . . . . . . . . . . . . 14 ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ∧ [(2nd𝑎) / 𝑦]𝑧 = 𝐶))
20 sbcan 2879 . . . . . . . . . . . . . . . 16 ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ↔ ([(2nd𝑎) / 𝑦]𝑥𝐴[(2nd𝑎) / 𝑦]𝑦𝐵))
21 vex 2622 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ V
22 2ndexg 5921 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ V → (2nd𝑎) ∈ V)
2321, 22ax-mp 7 . . . . . . . . . . . . . . . . . 18 (2nd𝑎) ∈ V
24 sbcg 2906 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑥𝐴𝑥𝐴))
2523, 24ax-mp 7 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑥𝐴𝑥𝐴)
26 sbcel1v 2899 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑦𝐵 ↔ (2nd𝑎) ∈ 𝐵)
2725, 26anbi12i 448 . . . . . . . . . . . . . . . 16 (([(2nd𝑎) / 𝑦]𝑥𝐴[(2nd𝑎) / 𝑦]𝑦𝐵) ↔ (𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵))
2820, 27bitri 182 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵))
29 sbceq2g 2951 . . . . . . . . . . . . . . . 16 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑧 = 𝐶𝑧 = (2nd𝑎) / 𝑦𝐶))
3023, 29ax-mp 7 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦]𝑧 = 𝐶𝑧 = (2nd𝑎) / 𝑦𝐶)
3128, 30anbi12i 448 . . . . . . . . . . . . . 14 (([(2nd𝑎) / 𝑦](𝑥𝐴𝑦𝐵) ∧ [(2nd𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
3219, 31bitri 182 . . . . . . . . . . . . 13 ([(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
3332sbcbii 2896 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ [(1st𝑎) / 𝑥]((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶))
34 sbcan 2879 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥]((𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (2nd𝑎) / 𝑦𝐶) ↔ ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ [(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶))
35 sbcan 2879 . . . . . . . . . . . . . 14 ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ↔ ([(1st𝑎) / 𝑥]𝑥𝐴[(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵))
36 sbcel1v 2899 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥]𝑥𝐴 ↔ (1st𝑎) ∈ 𝐴)
37 1stexg 5920 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ V → (1st𝑎) ∈ V)
3821, 37ax-mp 7 . . . . . . . . . . . . . . . 16 (1st𝑎) ∈ V
39 sbcg 2906 . . . . . . . . . . . . . . . 16 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵 ↔ (2nd𝑎) ∈ 𝐵))
4038, 39ax-mp 7 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵 ↔ (2nd𝑎) ∈ 𝐵)
4136, 40anbi12i 448 . . . . . . . . . . . . . 14 (([(1st𝑎) / 𝑥]𝑥𝐴[(1st𝑎) / 𝑥](2nd𝑎) ∈ 𝐵) ↔ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵))
4235, 41bitri 182 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ↔ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵))
43 sbceq2g 2951 . . . . . . . . . . . . . 14 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
4438, 43ax-mp 7 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
4542, 44anbi12i 448 . . . . . . . . . . . 12 (([(1st𝑎) / 𝑥](𝑥𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ [(1st𝑎) / 𝑥]𝑧 = (2nd𝑎) / 𝑦𝐶) ↔ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
4633, 34, 453bitri 204 . . . . . . . . . . 11 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦]((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶))
47 sbcan 2879 . . . . . . . . . . . . . 14 ([(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ ([(2nd𝑎) / 𝑦]𝑧𝐷[(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽)))
48 sbcg 2906 . . . . . . . . . . . . . . . 16 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑧𝐷𝑧𝐷))
4923, 48ax-mp 7 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦]𝑧𝐷𝑧𝐷)
50 sbcan 2879 . . . . . . . . . . . . . . . 16 ([(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽) ↔ ([(2nd𝑎) / 𝑦]𝑥 = 𝐼[(2nd𝑎) / 𝑦]𝑦 = 𝐽))
51 sbcg 2906 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑥 = 𝐼𝑥 = 𝐼))
5223, 51ax-mp 7 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑥 = 𝐼𝑥 = 𝐼)
53 sbceq1g 2949 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑎) ∈ V → ([(2nd𝑎) / 𝑦]𝑦 = 𝐽(2nd𝑎) / 𝑦𝑦 = 𝐽))
5423, 53ax-mp 7 . . . . . . . . . . . . . . . . . 18 ([(2nd𝑎) / 𝑦]𝑦 = 𝐽(2nd𝑎) / 𝑦𝑦 = 𝐽)
55 csbvarg 2956 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑎) ∈ V → (2nd𝑎) / 𝑦𝑦 = (2nd𝑎))
5623, 55ax-mp 7 . . . . . . . . . . . . . . . . . . 19 (2nd𝑎) / 𝑦𝑦 = (2nd𝑎)
5756eqeq1i 2095 . . . . . . . . . . . . . . . . . 18 ((2nd𝑎) / 𝑦𝑦 = 𝐽 ↔ (2nd𝑎) = 𝐽)
5854, 57bitri 182 . . . . . . . . . . . . . . . . 17 ([(2nd𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2nd𝑎) = 𝐽)
5952, 58anbi12i 448 . . . . . . . . . . . . . . . 16 (([(2nd𝑎) / 𝑦]𝑥 = 𝐼[(2nd𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽))
6050, 59bitri 182 . . . . . . . . . . . . . . 15 ([(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽))
6149, 60anbi12i 448 . . . . . . . . . . . . . 14 (([(2nd𝑎) / 𝑦]𝑧𝐷[(2nd𝑎) / 𝑦](𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
6247, 61bitri 182 . . . . . . . . . . . . 13 ([(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
6362sbcbii 2896 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ [(1st𝑎) / 𝑥](𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
64 sbcan 2879 . . . . . . . . . . . 12 ([(1st𝑎) / 𝑥](𝑧𝐷 ∧ (𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)) ↔ ([(1st𝑎) / 𝑥]𝑧𝐷[(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)))
65 sbcg 2906 . . . . . . . . . . . . . 14 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑧𝐷𝑧𝐷))
6638, 65ax-mp 7 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥]𝑧𝐷𝑧𝐷)
67 sbcan 2879 . . . . . . . . . . . . . 14 ([(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽) ↔ ([(1st𝑎) / 𝑥]𝑥 = 𝐼[(1st𝑎) / 𝑥](2nd𝑎) = 𝐽))
68 sbceq1g 2949 . . . . . . . . . . . . . . . . 17 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥]𝑥 = 𝐼(1st𝑎) / 𝑥𝑥 = 𝐼))
6938, 68ax-mp 7 . . . . . . . . . . . . . . . 16 ([(1st𝑎) / 𝑥]𝑥 = 𝐼(1st𝑎) / 𝑥𝑥 = 𝐼)
70 csbvarg 2956 . . . . . . . . . . . . . . . . . 18 ((1st𝑎) ∈ V → (1st𝑎) / 𝑥𝑥 = (1st𝑎))
7138, 70ax-mp 7 . . . . . . . . . . . . . . . . 17 (1st𝑎) / 𝑥𝑥 = (1st𝑎)
7271eqeq1i 2095 . . . . . . . . . . . . . . . 16 ((1st𝑎) / 𝑥𝑥 = 𝐼 ↔ (1st𝑎) = 𝐼)
7369, 72bitri 182 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1st𝑎) = 𝐼)
74 sbcg 2906 . . . . . . . . . . . . . . . 16 ((1st𝑎) ∈ V → ([(1st𝑎) / 𝑥](2nd𝑎) = 𝐽 ↔ (2nd𝑎) = 𝐽))
7538, 74ax-mp 7 . . . . . . . . . . . . . . 15 ([(1st𝑎) / 𝑥](2nd𝑎) = 𝐽 ↔ (2nd𝑎) = 𝐽)
7673, 75anbi12i 448 . . . . . . . . . . . . . 14 (([(1st𝑎) / 𝑥]𝑥 = 𝐼[(1st𝑎) / 𝑥](2nd𝑎) = 𝐽) ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))
7767, 76bitri 182 . . . . . . . . . . . . 13 ([(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽) ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))
7866, 77anbi12i 448 . . . . . . . . . . . 12 (([(1st𝑎) / 𝑥]𝑧𝐷[(1st𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2nd𝑎) = 𝐽)) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
7963, 64, 783bitri 204 . . . . . . . . . . 11 ([(1st𝑎) / 𝑥][(2nd𝑎) / 𝑦](𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽)) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
8018, 46, 793bitr3g 220 . . . . . . . . . 10 (𝜑 → ((((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
8180anbi2d 452 . . . . . . . . 9 (𝜑 → ((𝑎 ∈ (V × V) ∧ (((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))))
8215, 81syl5bb 190 . . . . . . . 8 (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))))
83 xpss 4534 . . . . . . . . . . . 12 (𝑋 × 𝑌) ⊆ (V × V)
84 simprr 499 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 = ⟨𝐼, 𝐽⟩)
858adantrr 463 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
8684, 85eqeltrd 2164 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (𝑋 × 𝑌))
8783, 86sseldi 3021 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (V × V))
8887ex 113 . . . . . . . . . 10 (𝜑 → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) → 𝑎 ∈ (V × V)))
8988pm4.71rd 386 . . . . . . . . 9 (𝜑 → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩))))
90 eqop 5929 . . . . . . . . . . 11 (𝑎 ∈ (V × V) → (𝑎 = ⟨𝐼, 𝐽⟩ ↔ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽)))
9190anbi2d 452 . . . . . . . . . 10 (𝑎 ∈ (V × V) → ((𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
9291pm5.32i 442 . . . . . . . . 9 ((𝑎 ∈ (V × V) ∧ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))))
9389, 92syl6rbb 195 . . . . . . . 8 (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧𝐷 ∧ ((1st𝑎) = 𝐼 ∧ (2nd𝑎) = 𝐽))) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9482, 93bitrd 186 . . . . . . 7 (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ 𝐴 ∧ (2nd𝑎) ∈ 𝐵)) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9514, 94syl5bb 190 . . . . . 6 (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)))
9695opabbidv 3896 . . . . 5 (𝜑 → {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)})
97 df-mpt2 5639 . . . . . . . 8 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
983, 97eqtri 2108 . . . . . . 7 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9998cnveqi 4599 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
100 nfv 1466 . . . . . . . 8 𝑥 𝑎 ∈ (𝐴 × 𝐵)
101 nfcsb1v 2961 . . . . . . . . 9 𝑥(1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
102101nfeq2 2240 . . . . . . . 8 𝑥 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
103100, 102nfan 1502 . . . . . . 7 𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
104 nfv 1466 . . . . . . . 8 𝑦 𝑎 ∈ (𝐴 × 𝐵)
105 nfcv 2228 . . . . . . . . . 10 𝑦(1st𝑎)
106 nfcsb1v 2961 . . . . . . . . . 10 𝑦(2nd𝑎) / 𝑦𝐶
107105, 106nfcsb 2963 . . . . . . . . 9 𝑦(1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
108107nfeq2 2240 . . . . . . . 8 𝑦 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶
109104, 108nfan 1502 . . . . . . 7 𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)
110 eleq1 2150 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
111 opelxp 4457 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
112110, 111syl6bb 194 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵)))
113 csbopeq1a 5940 . . . . . . . . 9 (𝑎 = ⟨𝑥, 𝑦⟩ → (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶 = 𝐶)
114113eqeq2d 2099 . . . . . . . 8 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶𝑧 = 𝐶))
115112, 114anbi12d 457 . . . . . . 7 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)))
116 xpss 4534 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
117116sseli 3019 . . . . . . . 8 (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V))
118117adantr 270 . . . . . . 7 ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶) → 𝑎 ∈ (V × V))
119103, 109, 115, 118cnvoprab 5981 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)}
12099, 119eqtri 2108 . . . . 5 𝐹 = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = (1st𝑎) / 𝑥(2nd𝑎) / 𝑦𝐶)}
121 df-mpt 3893 . . . . 5 (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) = {⟨𝑧, 𝑎⟩ ∣ (𝑧𝐷𝑎 = ⟨𝐼, 𝐽⟩)}
12296, 120, 1213eqtr4g 2145 . . . 4 (𝜑𝐹 = (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩))
123122fneq1d 5090 . . 3 (𝜑 → (𝐹 Fn 𝐷 ↔ (𝑧𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷))
12412, 123mpbird 165 . 2 (𝜑𝐹 Fn 𝐷)
125 dff1o4 5245 . 2 (𝐹:(𝐴 × 𝐵)–1-1-onto𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐹 Fn 𝐷))
1265, 124, 125sylanbrc 408 1 (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  Vcvv 2619  [wsbc 2838  csb 2931  cop 3444  {copab 3890  cmpt 3891   × cxp 4426  ccnv 4427   Fn wfn 4997  1-1-ontowf1o 5001  cfv 5002  {coprab 5635  cmpt2 5636  1st c1st 5891  2nd c2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by:  oddpwdc  11245
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