Theorem rneq 5004

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 4949	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4978	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4780	. 2 |- ran A = dom `' A
4		df-rn 4780	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1402   `'ccnv 4768   dom cdm 4769   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  rneqi  5005  rneqd  5006  xpima1  5229  feq1  5511  foeq1  5606  ixpsnf1o  7008  imasex  13603  ausgrusgrien  16326  0uhgrsubgr  16420