Theorem imaeq2i 5105

Description: Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Hypothesis

Ref	Expression
imaeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
imaeq2i	⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵)

Proof of Theorem imaeq2i

Step	Hyp	Ref	Expression
1		imaeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		imaeq2 5103	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 “ 𝐴) = (𝐶 “ 𝐵)

Syntax hints:   = wceq 1398   “ cima 4758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4761  df-cnv 4763  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768

This theorem is referenced by:  cnvimarndm  5132  dmco  5277  fnimapr  5743  ssimaex  5744  imauni  5941  isoini2  5999  fsuppeq  6461  fsuppeqg  6462  uniqs  6841  fiintim  7205  fidcenumlemrks  7237  fidcenumlemr  7239  fcdmnn0supp  9569  fcdmnn0fsupp  9570  fcdmnn0suppg  9571  nn0supp  9573  ennnfonelem1  13247  ennnfonelemhf1o  13253  ghmeqker  14029  retopbas  15519  eupth2lembfi  16603