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Theorem ovmpt2g 5662
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovmpt2g.1 (𝑥 = 𝐴𝑅 = 𝐺)
ovmpt2g.2 (𝑦 = 𝐵𝐺 = 𝑆)
ovmpt2g.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2g ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpt2g
StepHypRef Expression
1 ovmpt2g.1 . . 3 (𝑥 = 𝐴𝑅 = 𝐺)
2 ovmpt2g.2 . . 3 (𝑦 = 𝐵𝐺 = 𝑆)
31, 2sylan9eq 2108 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
4 ovmpt2g.3 . 2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
53, 4ovmpt2ga 5657 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 896   = wceq 1259  wcel 1409  (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:  ovmpt2  5663  oav  6064  omv  6065  oeiv  6066  mulpipq2  6526  genipv  6664  genpelxp  6666  subval  7265  divvalap  7726  cnref1o  8679  modqval  9273  frecuzrdgrrn  9357  frec2uzrdg  9358  frecuzrdgsuc  9364  iseqval  9383  iseqp1  9388  expival  9421  bcval  9616  shftfvalg  9646  shftfval  9649  cnrecnv  9737
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