Theorem rncoeq 5012

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 5011	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2233	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4742	. . . 4 |- ran B = dom `' B
4		dfdm4 4929	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2245	. . 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 4742	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4921	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4938	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2252	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4742	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2245	. 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 1398   `'ccnv 4730   dom cdm 4731   ran crn 4732   o. ccom 4735

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742

This theorem is referenced by:  dfdm2  5278  foco  5579