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Theorem imaeq12d 5625
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 5623 . 2 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
3 imaeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
43imaeq2d 5624 . 2 (𝜑 → (𝐵𝐶) = (𝐵𝐷))
52, 4eqtrd 2794 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cima 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  csbima12  5641  predeq123  5842  vdwpc  15906  dmdprd  18617  isunit  18877  qtopval  21720  limciun  23877  ig1pval  24151  ispth  26850  qqhval  30348  eulerpartgbij  30764  orvcval  30849  ballotlemrval  30909  ballotlemrinv0  30924  ballotlemrinv  30925  mthmval  31800  bj-projeq  33304  itg2addnclem2  33793  islmodfg  38159  heeq12  38590
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