Theorem ov2gf 7517

Description: The value of an operation class abstraction. A version of ovmpog 7527 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 3463	. . 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 2916	. . . . 5 ⊢ Ⅎ𝑥 𝐺 ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 7448	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
10		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 7398	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2913	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺
13	6, 12	nfim 1898	. . . 4 ⊢ Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14		ov2gf.2	. . . . . 6 ⊢ Ⅎ𝑦𝑆
15	14	nfel1 2916	. . . . 5 ⊢ Ⅎ𝑦 𝑆 ∈ V
16		nfmpo2 7449	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 7398	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2913	. . . . 5 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆
20	15, 19	nfim 1898	. . . 4 ⊢ Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21		ov2gf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)
22	21	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23		oveq1 7375	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
25	22, 24	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26		ov2gf.4	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)
27	26	eleq1d 2822	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28		oveq2 7376	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2753	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
30	27, 29	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
31	7	ovmpt4g 7515	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1122	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 3536	. . 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 1118	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  Vcvv 3442  (class class class)co 7368   ∈ cmpo 7370

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by: (None)