Theorem ov2gf 6145

Description: The value of an operation class abstraction. A version of ovmpog 6155 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 2814	. . 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 2385	. . . . 5 ⊢ Ⅎ𝑥 𝐺 ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 6087	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2371	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
10		nfcv 2374	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 6047	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2382	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺
13	6, 12	nfim 1620	. . . 4 ⊢ Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14		ov2gf.2	. . . . . 6 ⊢ Ⅎ𝑦𝑆
15	14	nfel1 2385	. . . . 5 ⊢ Ⅎ𝑦 𝑆 ∈ V
16		nfmpo2 6088	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2371	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 6047	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2382	. . . . 5 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆
20	15, 19	nfim 1620	. . . 4 ⊢ Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21		ov2gf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)
22	21	eleq1d 2300	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23		oveq1 6024	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2246	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
25	22, 24	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26		ov2gf.4	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆)
27	26	eleq1d 2300	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28		oveq2 6025	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2246	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
30	27, 29	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
31	7	ovmpt4g 6143	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1231	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2871	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))
34	1, 33	syl5 32	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝑆 ∈ 𝐻 → (𝐴𝐹𝐵) = 𝑆))
35	34	3impia 1226	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  Ⅎwnfc 2361  Vcvv 2802  (class class class)co 6017   ∈ cmpo 6019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022

This theorem is referenced by: (None)