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Theorem ov2gf 7582
Description: The value of an operation class abstraction. A version of ovmpog 7592 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 3499 . . 3 (𝑆𝐻𝑆 ∈ V)
2 ov2gf.a . . . 4 𝑥𝐴
3 ov2gf.c . . . 4 𝑦𝐴
4 ov2gf.d . . . 4 𝑦𝐵
5 ov2gf.1 . . . . . 6 𝑥𝐺
65nfel1 2920 . . . . 5 𝑥 𝐺 ∈ V
7 ov2gf.5 . . . . . . . 8 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpo1 7513 . . . . . . . 8 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2901 . . . . . . 7 𝑥𝐹
10 nfcv 2903 . . . . . . 7 𝑥𝑦
112, 9, 10nfov 7461 . . . . . 6 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2917 . . . . 5 𝑥(𝐴𝐹𝑦) = 𝐺
136, 12nfim 1894 . . . 4 𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14 ov2gf.2 . . . . . 6 𝑦𝑆
1514nfel1 2920 . . . . 5 𝑦 𝑆 ∈ V
16 nfmpo2 7514 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2901 . . . . . . 7 𝑦𝐹
183, 17, 4nfov 7461 . . . . . 6 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2917 . . . . 5 𝑦(𝐴𝐹𝐵) = 𝑆
2015, 19nfim 1894 . . . 4 𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21 ov2gf.3 . . . . . 6 (𝑥 = 𝐴𝑅 = 𝐺)
2221eleq1d 2824 . . . . 5 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23 oveq1 7438 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2751 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
2522, 24imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26 ov2gf.4 . . . . . 6 (𝑦 = 𝐵𝐺 = 𝑆)
2726eleq1d 2824 . . . . 5 (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28 oveq2 7439 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2751 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
3027, 29imbi12d 344 . . . 4 (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
317ovmpt4g 7580 . . . . 5 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1120 . . . 4 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3579 . . 3 ((𝐴𝐶𝐵𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))
341, 33syl5 34 . 2 ((𝐴𝐶𝐵𝐷) → (𝑆𝐻 → (𝐴𝐹𝐵) = 𝑆))
35343impia 1116 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wnfc 2888  Vcvv 3478  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by: (None)
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