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Theorem ovmpt2 5664
 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
ovmpt2g.1 (𝑥 = 𝐴𝑅 = 𝐺)
ovmpt2g.2 (𝑦 = 𝐵𝐺 = 𝑆)
ovmpt2g.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpt2.4 𝑆 ∈ V
Assertion
Ref Expression
ovmpt2 ((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 ovmpt2
StepHypRef Expression
1 ovmpt2.4 . 2 𝑆 ∈ V
2 ovmpt2g.1 . . 3 (𝑥 = 𝐴𝑅 = 𝐺)
3 ovmpt2g.2 . . 3 (𝑦 = 𝐵𝐺 = 𝑆)
4 ovmpt2g.3 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
52, 3, 4ovmpt2g 5663 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
61, 5mp3an3 1232 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  Vcvv 2574  (class class class)co 5540   ↦ cmpt2 5542 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 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290 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 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545 This theorem is referenced by:  ixxval  8866  fzval  8978
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