Theorem imaeq2d 4595

Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Hypothesis

Ref	Expression
imaeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
imaeq2d	⊢ (φ → (𝐶 “ A) = (𝐶 “ B))

Proof of Theorem imaeq2d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 ⊢ (φ → A = B)
2		imaeq2 4591	. 2 ⊢ (A = B → (𝐶 “ A) = (𝐶 “ B))
3	1, 2	syl 14	1 ⊢ (φ → (𝐶 “ A) = (𝐶 “ B))

Syntax hints:   → wi 4   = wceq 1228   “ cima 4275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285

This theorem is referenced by:  imaeq12d  4596  nfimad  4604  elimasng  4620  ressn  4785  foima  5036  f1imacnv  5068  fvco2  5167  fsn2  5262  resfunexg  5307  funfvima3  5317  funiunfvdm  5327  isoselem  5384  fnexALT  5663  eceq1  6052  uniqs2  6077  ecinxp  6092