Theorem ov2gf 7422

Description: The value of an operation class abstraction. A version of ovmpog 7432 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ov2gf.a	⊢ Ⅎ𝑥𝐴
ov2gf.c	⊢ Ⅎ𝑦𝐴
ov2gf.d	⊢ Ⅎ𝑦𝐵
ov2gf.1	⊢ Ⅎ𝑥𝐺
ov2gf.2	⊢ Ⅎ𝑦𝑆
ov2gf.3	⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)
ov2gf.4	⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)
ov2gf.5	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
ov2gf	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ov2gf

Step	Hyp	Ref	Expression
1		elex 3450	. . 3 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V)
2		ov2gf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		ov2gf.c	. . . 4 ⊢ Ⅎ𝑦𝐴
4		ov2gf.d	. . . 4 ⊢ Ⅎ𝑦𝐵
5		ov2gf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐺
6	5	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑥 𝐺 ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 7355	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
10		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 7305	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2920	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺
13	6, 12	nfim 1899	. . . 4 ⊢ Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14		ov2gf.2	. . . . . 6 ⊢ Ⅎ𝑦𝑆
15	14	nfel1 2923	. . . . 5 ⊢ Ⅎ𝑦 𝑆 ∈ V
16		nfmpo2 7356	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 7305	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2920	. . . . 5 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆
20	15, 19	nfim 1899	. . . 4 ⊢ Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21		ov2gf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)
22	21	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23		oveq1 7282	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
25	22, 24	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26		ov2gf.4	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)
27	26	eleq1d 2823	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28		oveq2 7283	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2754	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
30	27, 29	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
31	7	ovmpt4g 7420	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1120	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 3515	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))
34	1, 33	syl5 34	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ 𝐻 → (𝐴𝐹𝐵) = 𝑆))
35	34	3impia 1116	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  Vcvv 3432  (class class class)co 7275   ∈ cmpo 7277

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280

This theorem is referenced by: (None)