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

Theorem imaeq12d 6018
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 6016 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 imaeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43imaeq2d 6017 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2773 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  csbima12  6035  predeq123  6258  vdwpc  16860  dmdprd  19785  isunit  20094  qtopval  23069  limciun  25281  ig1pval  25560  ispth  28720  irngval  32423  qqhval  32619  eulerpartgbij  33036  orvcval  33121  ballotlemrval  33181  ballotlemrinv0  33196  ballotlemrinv  33197  mthmval  34233  bj-projeq  35513  itg2addnclem2  36180  islmodfg  41443  heeq12  42140
  Copyright terms: Public domain W3C validator