Theorem imaeq1 2602

Description: Equality theorem for image.

Assertion

Ref	Expression
imaeq1	|- (A = B -> (A"C) = (B"C))

Proof of Theorem imaeq1

Step	Hyp	Ref	Expression
1		reseq1 2575	. . 3 |- (A = B -> (A |` C) = (B |` C))
2	1	rneqd 2557	. 2 |- (A = B -> ran (A |` C) = ran (B |` C))
3		df-ima 2431	. 2 |- (A"C) = ran (A |` C)
4		df-ima 2431	. 2 |- (B"C) = ran (B |` C)
5	2, 3, 4	3eqtr4g 1147	1 |- (A = B -> (A"C) = (B"C))

Syntax hints:   -> wi 2   = wceq 1091  ran crn 2411   |` cres 2412  "cima 2413

This theorem is referenced by:  f1imacnv 2814  fveq1 2831  eceq1 3214  ssenen 3399

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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