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Theorem imaeq1 4844
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 4781 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
21rneqd 4736 . 2 (𝐴 = 𝐵 → ran (𝐴𝐶) = ran (𝐵𝐶))
3 df-ima 4520 . 2 (𝐴𝐶) = ran (𝐴𝐶)
4 df-ima 4520 . 2 (𝐵𝐶) = ran (𝐵𝐶)
52, 3, 43eqtr4g 2173 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  ran crn 4508  cres 4509  cima 4510
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  imaeq1i  4846  imaeq1d  4848  eceq2  6432  iscnp  12263
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