Theorem rncoeq 4807

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 4806	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2139	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4545	. . . 4 |- ran B = dom `' B
4		dfdm4 4726	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2151	. . 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 4545	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4719	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4735	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2158	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4545	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2151	. 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 1331   `'ccnv 4533   dom cdm 4534   ran crn 4535   o. ccom 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545

This theorem is referenced by:  dfdm2  5068  foco  5350