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Theorem imaeq2d 4953
Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
Hypothesis
Ref Expression
imaeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
imaeq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem imaeq2d
StepHypRef Expression
1 imaeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 imaeq2 4949 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  cima 4614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624
This theorem is referenced by:  imaeq12d  4954  nfimad  4962  elimasng  4979  ressn  5151  foima  5425  f1imacnv  5459  fvco2  5565  fsn2  5670  resfunexg  5717  funfvima3  5729  funiunfvdm  5742  isoselem  5799  fnexALT  6090  eceq1  6548  uniqs2  6573  ecinxp  6588  mapsn  6668  phplem4  6833  phplem4dom  6840  phplem4on  6845  sbthlem2  6935  isbth  6944  resunimafz0  10766  ennnfonelemg  12358  ennnfonelemhf1o  12368  ennnfonelemex  12369  ennnfonelemrn  12374  cnntr  13019  cnptopresti  13032  cnptoprest  13033
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