Theorem rncoeq 4605

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 4604	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2042	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4356	. . . 4 |- ran B = dom `' B
4		dfdm4 4527	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2053	. . 3 |- ( ran B = dom A <-> dom `' B = ran `' A )
6	2, 5	bitri 173	. 2 |- ( dom A = ran B <-> dom `' B = ran `' A )
7		df-rn 4356	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4520	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4536	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2060	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4356	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2053	. 2 |- ( ran ( A o. B ) = ran A <-> dom ( `' B o. `' A ) = dom `' A )
13	1, 6, 12	3imtr4i 190	1 |- ( dom A = ran B -> ran ( A o. B ) = ran A )

Syntax hints:   -> wi 4   = wceq 1243   `'ccnv 4344   dom cdm 4345   ran crn 4346   o. ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356

This theorem is referenced by:  dfdm2  4852  foco  5116