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Theorem ovmpo 5872
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovmpog.1 (𝑥 = 𝐴𝑅 = 𝐺)
ovmpog.2 (𝑦 = 𝐵𝐺 = 𝑆)
ovmpog.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpo.4 𝑆 ∈ V
Assertion
Ref Expression
ovmpo ((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)   𝐺(𝑥,𝑦)

Proof of Theorem ovmpo
StepHypRef Expression
1 ovmpo.4 . 2 𝑆 ∈ V
2 ovmpog.1 . . 3 (𝑥 = 𝐴𝑅 = 𝐺)
3 ovmpog.2 . . 3 (𝑦 = 𝐵𝐺 = 𝑆)
4 ovmpog.3 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
52, 3, 4ovmpog 5871 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
61, 5mp3an3 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:  ixxval  9619  fzval  9732
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