f1od2 - Mathbox for Thierry Arnoux

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Theorem f1od2 29499

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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
f1od2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)
f1od2.3	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))
f1od2.4	⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))

**Assertion**
Ref	Expression
f1od2	⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

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.

Step	Hyp	Ref	Expression
1		f1od2.2	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)
2	1	ralrimivva 2971	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊)
3		f1od2.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	3	fnmpt2 7238	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑊 → 𝐹 Fn (𝐴 × 𝐵))
5	2, 4	syl 17	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))
6		f1od2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))
7		opelxpi 5148	. . . . . 6 ⊢ ((𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
8	6, 7	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
9	8	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
10		eqid 2622	. . . . 5 ⊢ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) = (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩)
11	10	fnmpt 6020	. . . 4 ⊢ (∀𝑧 ∈ 𝐷 ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌) → (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷)
13		elxp7 7201	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ (V × V) ∧ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵)))
14	13	anbi1i 731	. . . . . . 7 ⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑎 ∈ (V × V) ∧ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
15		anass 681	. . . . . . . . 9 ⊢ (((𝑎 ∈ (V × V) ∧ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)))
16		f1od2.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
17	16	sbcbidv 3490	. . . . . . . . . . . 12 ⊢ (𝜑 → ([(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
18	17	sbcbidv 3490	. . . . . . . . . . 11 ⊢ (𝜑 → ([(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))
19		sbcan 3478	. . . . . . . . . . . . . 14 ⊢ ([(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ([(2^nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2^nd ‘𝑎) / 𝑦]𝑧 = 𝐶))
20		sbcan 3478	. . . . . . . . . . . . . . . 16 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ([(2^nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2^nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵))
21		fvex 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘𝑎) ∈ V
22		sbcg 3503	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑎) ∈ V → ([(2^nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
24		sbcel1v 3495	. . . . . . . . . . . . . . . . 17 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵 ↔ (2^nd ‘𝑎) ∈ 𝐵)
25	23, 24	anbi12i 733	. . . . . . . . . . . . . . . 16 ⊢ (([(2^nd ‘𝑎) / 𝑦]𝑥 ∈ 𝐴 ∧ [(2^nd ‘𝑎) / 𝑦]𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵))
26	20, 25	bitri 264	. . . . . . . . . . . . . . 15 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵))
27		sbceq2g 3990	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑎) ∈ V → ([(2^nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
28	21, 27	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑧 = 𝐶 ↔ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)
29	26, 28	anbi12i 733	. . . . . . . . . . . . . 14 ⊢ (([(2^nd ‘𝑎) / 𝑦](𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(2^nd ‘𝑎) / 𝑦]𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
30	19, 29	bitri 264	. . . . . . . . . . . . 13 ⊢ ([(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
31	30	sbcbii 3491	. . . . . . . . . . . 12 ⊢ ([(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ [(1^st ‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
32		sbcan 3478	. . . . . . . . . . . 12 ⊢ ([(1^st ‘𝑎) / 𝑥]((𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ ([(1^st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ [(1^st ‘𝑎) / 𝑥]𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
33		sbcan 3478	. . . . . . . . . . . . . 14 ⊢ ([(1^st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ↔ ([(1^st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) ∈ 𝐵))
34		sbcel1v 3495	. . . . . . . . . . . . . . 15 ⊢ ([(1^st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ↔ (1^st ‘𝑎) ∈ 𝐴)
35		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘𝑎) ∈ V
36		sbcg 3503	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑎) ∈ V → ([(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) ∈ 𝐵 ↔ (2^nd ‘𝑎) ∈ 𝐵))
37	35, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) ∈ 𝐵 ↔ (2^nd ‘𝑎) ∈ 𝐵)
38	34, 37	anbi12i 733	. . . . . . . . . . . . . 14 ⊢ (([(1^st ‘𝑎) / 𝑥]𝑥 ∈ 𝐴 ∧ [(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) ∈ 𝐵) ↔ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵))
39	33, 38	bitri 264	. . . . . . . . . . . . 13 ⊢ ([(1^st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ↔ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵))
40		sbceq2g 3990	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑎) ∈ V → ([(1^st ‘𝑎) / 𝑥]𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
41	35, 40	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([(1^st ‘𝑎) / 𝑥]𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)
42	39, 41	anbi12i 733	. . . . . . . . . . . 12 ⊢ (([(1^st ‘𝑎) / 𝑥](𝑥 ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ [(1^st ‘𝑎) / 𝑥]𝑧 = ⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
43	31, 32, 42	3bitri 286	. . . . . . . . . . 11 ⊢ ([(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦]((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶))
44		sbcan 3478	. . . . . . . . . . . . . 14 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ ([(2^nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2^nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))
45		sbcg 3503	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑎) ∈ V → ([(2^nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷))
46	21, 45	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)
47		sbcan 3478	. . . . . . . . . . . . . . . 16 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ ([(2^nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2^nd ‘𝑎) / 𝑦]𝑦 = 𝐽))
48		sbcg 3503	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘𝑎) ∈ V → ([(2^nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼))
49	21, 48	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ↔ 𝑥 = 𝐼)
50		sbceq1g 3988	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑎) ∈ V → ([(2^nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2^nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽))
51	21, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ ⦋(2^nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽)
52		csbvarg 4003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑎) ∈ V → ⦋(2^nd ‘𝑎) / 𝑦⦌𝑦 = (2^nd ‘𝑎))
53	21, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ⦋(2^nd ‘𝑎) / 𝑦⦌𝑦 = (2^nd ‘𝑎)
54	53	eqeq1i 2627	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋(2^nd ‘𝑎) / 𝑦⦌𝑦 = 𝐽 ↔ (2^nd ‘𝑎) = 𝐽)
55	51, 54	bitri 264	. . . . . . . . . . . . . . . . 17 ⊢ ([(2^nd ‘𝑎) / 𝑦]𝑦 = 𝐽 ↔ (2^nd ‘𝑎) = 𝐽)
56	49, 55	anbi12i 733	. . . . . . . . . . . . . . . 16 ⊢ (([(2^nd ‘𝑎) / 𝑦]𝑥 = 𝐼 ∧ [(2^nd ‘𝑎) / 𝑦]𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))
57	47, 56	bitri 264	. . . . . . . . . . . . . . 15 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))
58	46, 57	anbi12i 733	. . . . . . . . . . . . . 14 ⊢ (([(2^nd ‘𝑎) / 𝑦]𝑧 ∈ 𝐷 ∧ [(2^nd ‘𝑎) / 𝑦](𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
59	44, 58	bitri 264	. . . . . . . . . . . . 13 ⊢ ([(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
60	59	sbcbii 3491	. . . . . . . . . . . 12 ⊢ ([(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ [(1^st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
61		sbcan 3478	. . . . . . . . . . . 12 ⊢ ([(1^st ‘𝑎) / 𝑥](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)) ↔ ([(1^st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1^st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
62		sbcg 3503	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑎) ∈ V → ([(1^st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷))
63	35, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([(1^st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷)
64		sbcan 3478	. . . . . . . . . . . . . 14 ⊢ ([(1^st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽) ↔ ([(1^st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) = 𝐽))
65		sbceq1g 3988	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑎) ∈ V → ([(1^st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1^st ‘𝑎) / 𝑥⦌𝑥 = 𝐼))
66	35, 65	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ([(1^st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ ⦋(1^st ‘𝑎) / 𝑥⦌𝑥 = 𝐼)
67		csbvarg 4003	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑎) ∈ V → ⦋(1^st ‘𝑎) / 𝑥⦌𝑥 = (1^st ‘𝑎))
68	35, 67	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ⦋(1^st ‘𝑎) / 𝑥⦌𝑥 = (1^st ‘𝑎)
69	68	eqeq1i 2627	. . . . . . . . . . . . . . . 16 ⊢ (⦋(1^st ‘𝑎) / 𝑥⦌𝑥 = 𝐼 ↔ (1^st ‘𝑎) = 𝐼)
70	66, 69	bitri 264	. . . . . . . . . . . . . . 15 ⊢ ([(1^st ‘𝑎) / 𝑥]𝑥 = 𝐼 ↔ (1^st ‘𝑎) = 𝐼)
71		sbcg 3503	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑎) ∈ V → ([(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) = 𝐽 ↔ (2^nd ‘𝑎) = 𝐽))
72	35, 71	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ([(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) = 𝐽 ↔ (2^nd ‘𝑎) = 𝐽)
73	70, 72	anbi12i 733	. . . . . . . . . . . . . 14 ⊢ (([(1^st ‘𝑎) / 𝑥]𝑥 = 𝐼 ∧ [(1^st ‘𝑎) / 𝑥](2^nd ‘𝑎) = 𝐽) ↔ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))
74	64, 73	bitri 264	. . . . . . . . . . . . 13 ⊢ ([(1^st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽) ↔ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))
75	63, 74	anbi12i 733	. . . . . . . . . . . 12 ⊢ (([(1^st ‘𝑎) / 𝑥]𝑧 ∈ 𝐷 ∧ [(1^st ‘𝑎) / 𝑥](𝑥 = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
76	60, 61, 75	3bitri 286	. . . . . . . . . . 11 ⊢ ([(1^st ‘𝑎) / 𝑥][(2^nd ‘𝑎) / 𝑦](𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) ↔ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
77	18, 43, 76	3bitr3g 302	. . . . . . . . . 10 ⊢ (𝜑 → ((((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))))
78	77	anbi2d 740	. . . . . . . . 9 ⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))))
79	15, 78	syl5bb 272	. . . . . . . 8 ⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))))
80		xpss 5226	. . . . . . . . . . . 12 ⊢ (𝑋 × 𝑌) ⊆ (V × V)
81		simprr 796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 = ⟨𝐼, 𝐽⟩)
82	8	adantrr 753	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → ⟨𝐼, 𝐽⟩ ∈ (𝑋 × 𝑌))
83	81, 82	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (𝑋 × 𝑌))
84	80, 83	sseldi 3601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) → 𝑎 ∈ (V × V))
85	84	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) → 𝑎 ∈ (V × V)))
86	85	pm4.71rd 667	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩))))
87		eqop 7208	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (V × V) → (𝑎 = ⟨𝐼, 𝐽⟩ ↔ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽)))
88	87	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑎 ∈ (V × V) → ((𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩) ↔ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))))
89	88	pm5.32i 669	. . . . . . . . 9 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)) ↔ (𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))))
90	86, 89	syl6rbb 277	. . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (V × V) ∧ (𝑧 ∈ 𝐷 ∧ ((1^st ‘𝑎) = 𝐼 ∧ (2^nd ‘𝑎) = 𝐽))) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
91	79, 90	bitrd 268	. . . . . . 7 ⊢ (𝜑 → (((𝑎 ∈ (V × V) ∧ ((1^st ‘𝑎) ∈ 𝐴 ∧ (2^nd ‘𝑎) ∈ 𝐵)) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
92	14, 91	syl5bb 272	. . . . . 6 ⊢ (𝜑 → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)))
93	92	opabbidv 4716	. . . . 5 ⊢ (𝜑 → {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)})
94		df-mpt2 6655	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
95	3, 94	eqtri 2644	. . . . . . 7 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
96	95	cnveqi 5297	. . . . . 6 ⊢ ^◡𝐹 = ^◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
97		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ (𝐴 × 𝐵)
98		nfcsb1v 3549	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶
99	98	nfeq2 2780	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶
100	97, 99	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑥(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)
101		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑎 ∈ (𝐴 × 𝐵)
102		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑦(1^st ‘𝑎)
103		nfcsb1v 3549	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋(2^nd ‘𝑎) / 𝑦⦌𝐶
104	102, 103	nfcsb 3551	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶
105	104	nfeq2 2780	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶
106	101, 105	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑦(𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)
107		eleq1 2689	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
108		opelxp 5146	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
109	107, 108	syl6bb 276	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑎 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
110		csbopeq1a 7221	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶 = 𝐶)
111	110	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶 ↔ 𝑧 = 𝐶))
112	109, 111	anbi12d 747	. . . . . . 7 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)))
113		xpss 5226	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
114	113	sseli 3599	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐴 × 𝐵) → 𝑎 ∈ (V × V))
115	114	adantr 481	. . . . . . 7 ⊢ ((𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶) → 𝑎 ∈ (V × V))
116	100, 106, 112, 115	cnvoprab 29498	. . . . . 6 ⊢ ^◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)}
117	96, 116	eqtri 2644	. . . . 5 ⊢ ^◡𝐹 = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (𝐴 × 𝐵) ∧ 𝑧 = ⦋(1^st ‘𝑎) / 𝑥⦌⦋(2^nd ‘𝑎) / 𝑦⦌𝐶)}
118		df-mpt 4730	. . . . 5 ⊢ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) = {⟨𝑧, 𝑎⟩ ∣ (𝑧 ∈ 𝐷 ∧ 𝑎 = ⟨𝐼, 𝐽⟩)}
119	93, 117, 118	3eqtr4g 2681	. . . 4 ⊢ (𝜑 → ^◡𝐹 = (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩))
120	119	fneq1d 5981	. . 3 ⊢ (𝜑 → (^◡𝐹 Fn 𝐷 ↔ (𝑧 ∈ 𝐷 ↦ ⟨𝐼, 𝐽⟩) Fn 𝐷))
121	12, 120	mpbird 247	. 2 ⊢ (𝜑 → ^◡𝐹 Fn 𝐷)
122		dff1o4 6145	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ^◡𝐹 Fn 𝐷))
123	5, 121, 122	sylanbrc 698	1 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 [wsbc 3435 ⦋csb 3533 ⟨cop 4183 {copab 4712 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 Fn wfn 5883 –1-1-onto→wf1o 5887 ‘cfv 5888 {coprab 6651 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169

This theorem is referenced by: oddpwdc 30416

W3C validator