MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2ga Structured version   Visualization version   GIF version

Theorem ovmpt2ga 6941
Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2ga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2ga.2 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2ga ((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)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpt2ga
StepHypRef Expression
1 elex 3364 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpt2ga.2 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 11 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpt2ga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 467 . . 3 (((𝐴𝐶𝐵𝐷𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 simp1 1130 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐴𝐶)
7 simp2 1131 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝐵𝐷)
8 simp3 1132 . . 3 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → 𝑆 ∈ V)
93, 5, 6, 7, 8ovmpt2d 6939 . 2 ((𝐴𝐶𝐵𝐷𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
101, 9syl3an3 1169 1 ((𝐴𝐶𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  (class class class)co 6796  cmpt2 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801
This theorem is referenced by:  ovmpt2a  6942  ovmpt2g  6946  elovmpt2  7030  offval  7055  offval3  7313  mptmpt2opabbrd  7402  bropopvvv  7410  reps  13726  hashbcval  15913  setsvalg  16094  ressval  16134  restval  16295  sylow1lem4  18223  sylow3lem2  18250  sylow3lem3  18251  lsmvalx  18261  mvrfval  19635  opsrval  19689  marrepfval  20584  marrepval0  20585  marepvfval  20589  marepvval0  20590  cnmpt12  21691  cnmpt22  21698  qtopval  21719  flimval  21987  fclsval  22032  ucnval  22301  stdbdmetval  22539  resvval  30167  ofcfval3  30504  fmulcl  40326
  Copyright terms: Public domain W3C validator