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Theorem imaeq12d 6061
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 6059 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 imaeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43imaeq2d 6060 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2773 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cima 5680
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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  csbima12  6079  predeq123  6302  vdwpc  16913  dmdprd  19868  isunit  20187  qtopval  23199  limciun  25411  ig1pval  25690  ispth  28980  irngval  32749  qqhval  32954  eulerpartgbij  33371  orvcval  33456  ballotlemrval  33516  ballotlemrinv0  33531  ballotlemrinv  33532  mthmval  34566  bj-projeq  35873  itg2addnclem2  36540  islmodfg  41811  heeq12  42527
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