Theorem ov2gf 6178

Description: The value of an operation class abstraction. A version of ovmpog 6188 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 2825	. . 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 2395	. . . . 5 ⊢ Ⅎ𝑥 𝐺 ∈ V
7		ov2gf.5	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 6120	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2381	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
10		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 6080	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2392	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = 𝐺
13	6, 12	nfim 1621	. . . 4 ⊢ Ⅎ𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14		ov2gf.2	. . . . . 6 ⊢ Ⅎ𝑦𝑆
15	14	nfel1 2395	. . . . 5 ⊢ Ⅎ𝑦 𝑆 ∈ V
16		nfmpo2 6121	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2381	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 6080	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2392	. . . . 5 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = 𝑆
20	15, 19	nfim 1621	. . . 4 ⊢ Ⅎ𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21		ov2gf.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺)
22	21	eleq1d 2301	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23		oveq1 6057	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2247	. . . . 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 2301	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28		oveq2 6058	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2247	. . . . 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 6176	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1232	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2882	. . 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 1227	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Ⅎwnfc 2371  Vcvv 2813  (class class class)co 6050   ∈ cmpo 6052

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055

This theorem is referenced by: (None)