Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaeq12d Structured version   Visualization version   GIF version

Theorem imaeq12d 5908
 Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
Hypotheses
Ref Expression
imaeq1d.1 (𝜑𝐴 = 𝐵)
imaeq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
imaeq12d (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem imaeq12d
StepHypRef Expression
1 imaeq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21imaeq1d 5906 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 imaeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43imaeq2d 5907 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2857 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   “ cima 5535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545 This theorem is referenced by:  csbima12  5925  predeq123  6127  vdwpc  16305  dmdprd  19111  isunit  19401  qtopval  22298  limciun  24495  ig1pval  24771  ispth  27510  qqhval  31289  eulerpartgbij  31704  orvcval  31789  ballotlemrval  31849  ballotlemrinv0  31864  ballotlemrinv  31865  mthmval  32896  bj-projeq  34389  itg2addnclem2  35067  islmodfg  39943  heeq12  40408
 Copyright terms: Public domain W3C validator