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Theorem ovmpt2a 5757
 Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2ga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2ga.2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
ovmpt2a.4 𝑆 ∈ V
Assertion
Ref Expression
ovmpt2a ((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 ovmpt2a
StepHypRef Expression
1 ovmpt2a.4 . 2 𝑆 ∈ V
2 ovmpt2ga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
3 ovmpt2ga.2 . . 3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
42, 3ovmpt2ga 5756 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
51, 4mp3an3 1262 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝐹𝐵) = 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∈ wcel 1438  Vcvv 2619  (class class class)co 5634   ↦ cmpt2 5636 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343 This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639 This theorem is referenced by: (None)
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