Theorem rncoeq 4877

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 4876	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2167	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4615	. . . 4 |- ran B = dom `' B
4		dfdm4 4796	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2179	. . 3 |- ( ran B = dom A <-> dom `' B = ran `' A )
6	2, 5	bitri 183	. 2 |- ( dom A = ran B <-> dom `' B = ran `' A )
7		df-rn 4615	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4789	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4805	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2186	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4615	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2179	. 2 |- ( ran ( A o. B ) = ran A <-> dom ( `' B o. `' A ) = dom `' A )
13	1, 6, 12	3imtr4i 200	1 |- ( dom A = ran B -> ran ( A o. B ) = ran A )

Syntax hints:   -> wi 4   = wceq 1343   `'ccnv 4603   dom cdm 4604   ran crn 4605   o. ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615

This theorem is referenced by:  dfdm2  5138  foco  5420