Theorem rneq 5009

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 4954	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4983	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4785	. 2 |- ran A = dom `' A
4		df-rn 4785	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ran A = ran B )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   `'ccnv 4773   dom cdm 4774   ran crn 4775

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  rneqi  5010  rneqd  5011  xpima1  5234  feq1  5516  foeq1  5611  ixpsnf1o  7018  imasex  13626  ausgrusgrien  16412  0uhgrsubgr  16506