Theorem rneq 4905

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
rneq	|- ( A = B -> ran A = ran B )

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		cnveq 4852	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4880	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4686	. 2 |- ran A = dom `' A
4		df-rn 4686	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4674   dom cdm 4675   ran crn 4676

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  rneqi  4906  rneqd  4907  xpima1  5129  feq1  5408  foeq1  5494  ixpsnf1o  6823  imasex  13137