Theorem rneq 4914

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 4860	. . 3 |- ( A = B -> `' A = `' B )
2	1	dmeqd 4889	. 2 |- ( A = B -> dom `' A = dom `' B )
3		df-rn 4694	. 2 |- ran A = dom `' A
4		df-rn 4694	. 2 |- ran B = dom `' B
5	2, 3, 4	3eqtr4g 2264	1 |- ( A = B -> ran A = ran B )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4682   dom cdm 4683   ran crn 4684

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-cnv 4691  df-dm 4693  df-rn 4694

This theorem is referenced by:  rneqi  4915  rneqd  4916  xpima1  5138  feq1  5418  foeq1  5506  ixpsnf1o  6836  imasex  13212