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Theorem ov2gf 6738
Description: The value of an operation class abstraction. A version of ovmpt2g 6748 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ov2gf.a 𝑥𝐴
ov2gf.c 𝑦𝐴
ov2gf.d 𝑦𝐵
ov2gf.1 𝑥𝐺
ov2gf.2 𝑦𝑆
ov2gf.3 (𝑥 = 𝐴𝑅 = 𝐺)
ov2gf.4 (𝑦 = 𝐵𝐺 = 𝑆)
ov2gf.5 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ov2gf ((𝐴𝐶𝐵𝐷𝑆𝐻) → (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
StepHypRef Expression
1 elex 3198 . . 3 (𝑆𝐻𝑆 ∈ V)
2 ov2gf.a . . . 4 𝑥𝐴
3 ov2gf.c . . . 4 𝑦𝐴
4 ov2gf.d . . . 4 𝑦𝐵
5 ov2gf.1 . . . . . 6 𝑥𝐺
65nfel1 2775 . . . . 5 𝑥 𝐺 ∈ V
7 ov2gf.5 . . . . . . . 8 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpt21 6675 . . . . . . . 8 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2759 . . . . . . 7 𝑥𝐹
10 nfcv 2761 . . . . . . 7 𝑥𝑦
112, 9, 10nfov 6630 . . . . . 6 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2772 . . . . 5 𝑥(𝐴𝐹𝑦) = 𝐺
136, 12nfim 1822 . . . 4 𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14 ov2gf.2 . . . . . 6 𝑦𝑆
1514nfel1 2775 . . . . 5 𝑦 𝑆 ∈ V
16 nfmpt22 6676 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2759 . . . . . . 7 𝑦𝐹
183, 17, 4nfov 6630 . . . . . 6 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2772 . . . . 5 𝑦(𝐴𝐹𝐵) = 𝑆
2015, 19nfim 1822 . . . 4 𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21 ov2gf.3 . . . . . 6 (𝑥 = 𝐴𝑅 = 𝐺)
2221eleq1d 2683 . . . . 5 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23 oveq1 6611 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2636 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
2522, 24imbi12d 334 . . . 4 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26 ov2gf.4 . . . . . 6 (𝑦 = 𝐵𝐺 = 𝑆)
2726eleq1d 2683 . . . . 5 (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28 oveq2 6612 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2636 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
3027, 29imbi12d 334 . . . 4 (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
317ovmpt4g 6736 . . . . 5 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1264 . . . 4 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3259 . . 3 ((𝐴𝐶𝐵𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))
341, 33syl5 34 . 2 ((𝐴𝐶𝐵𝐷) → (𝑆𝐻 → (𝐴𝐹𝐵) = 𝑆))
35343impia 1258 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wnfc 2748  Vcvv 3186  (class class class)co 6604  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609
This theorem is referenced by: (None)
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