Theorem rncoeq 4971

Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)

Assertion

Ref	Expression
rncoeq	|- ( dom A = ran B -> ran ( A o. B ) = ran A )

Proof of Theorem rncoeq

Step	Hyp	Ref	Expression
1		dmcoeq 4970	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2209	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4704	. . . 4 |- ran B = dom `' B
4		dfdm4 4889	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2221	. . 3 |- ( ran B = dom A <-> dom `' B = ran `' A )
6	2, 5	bitri 184	. 2 |- ( dom A = ran B <-> dom `' B = ran `' A )
7		df-rn 4704	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4881	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4898	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2228	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4704	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2221	. 2 |- ( ran ( A o. B ) = ran A <-> dom ( `' B o. `' A ) = dom `' A )
13	1, 6, 12	3imtr4i 201	1 |- ( dom A = ran B -> ran ( A o. B ) = ran A )

Syntax hints:   -> wi 4   = wceq 1373   `'ccnv 4692   dom cdm 4693   ran crn 4694   o. ccom 4697

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-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704

This theorem is referenced by:  dfdm2  5236  foco  5531