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Theorem ov2gf 5652
Description: The value of an operation class abstraction. A version of ovmpt2g 5662 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 2583 . . 3 (𝑆𝐻𝑆 ∈ V)
2 ov2gf.a . . . 4 𝑥𝐴
3 ov2gf.c . . . 4 𝑦𝐴
4 ov2gf.d . . . 4 𝑦𝐵
5 ov2gf.1 . . . . . 6 𝑥𝐺
65nfel1 2204 . . . . 5 𝑥 𝐺 ∈ V
7 ov2gf.5 . . . . . . . 8 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpt21 5598 . . . . . . . 8 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2191 . . . . . . 7 𝑥𝐹
10 nfcv 2194 . . . . . . 7 𝑥𝑦
112, 9, 10nfov 5562 . . . . . 6 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2201 . . . . 5 𝑥(𝐴𝐹𝑦) = 𝐺
136, 12nfim 1480 . . . 4 𝑥(𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)
14 ov2gf.2 . . . . . 6 𝑦𝑆
1514nfel1 2204 . . . . 5 𝑦 𝑆 ∈ V
16 nfmpt22 5599 . . . . . . . 8 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2191 . . . . . . 7 𝑦𝐹
183, 17, 4nfov 5562 . . . . . 6 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2201 . . . . 5 𝑦(𝐴𝐹𝐵) = 𝑆
2015, 19nfim 1480 . . . 4 𝑦(𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)
21 ov2gf.3 . . . . . 6 (𝑥 = 𝐴𝑅 = 𝐺)
2221eleq1d 2122 . . . . 5 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐺 ∈ V))
23 oveq1 5546 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2070 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐺))
2522, 24imbi12d 227 . . . 4 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺)))
26 ov2gf.4 . . . . . 6 (𝑦 = 𝐵𝐺 = 𝑆)
2726eleq1d 2122 . . . . 5 (𝑦 = 𝐵 → (𝐺 ∈ V ↔ 𝑆 ∈ V))
28 oveq2 5547 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2070 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐺 ↔ (𝐴𝐹𝐵) = 𝑆))
3027, 29imbi12d 227 . . . 4 (𝑦 = 𝐵 → ((𝐺 ∈ V → (𝐴𝐹𝑦) = 𝐺) ↔ (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆)))
317ovmpt4g 5650 . . . . 5 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1117 . . . 4 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2637 . . 3 ((𝐴𝐶𝐵𝐷) → (𝑆 ∈ V → (𝐴𝐹𝐵) = 𝑆))
341, 33syl5 32 . 2 ((𝐴𝐶𝐵𝐷) → (𝑆𝐻 → (𝐴𝐹𝐵) = 𝑆))
35343impia 1112 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  wnfc 2181  Vcvv 2574  (class class class)co 5539  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544
This theorem is referenced by: (None)
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