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Theorem ovmpoa 5867
Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpoga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpoga.2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpoa.4 𝑆 ∈ V
Assertion
Ref Expression
ovmpoa ((𝐴𝐶𝐵𝐷) → (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)

Proof of Theorem ovmpoa
StepHypRef Expression
1 ovmpoa.4 . 2 𝑆 ∈ V
2 ovmpoga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
3 ovmpoga.2 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
42, 3ovmpoga 5866 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
51, 4mp3an3 1287 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  Vcvv 2658  (class class class)co 5740  cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745
This theorem is referenced by: (None)
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