Theorem rncoeq 4871

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 4870	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2166	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4609	. . . 4 |- ran B = dom `' B
4		dfdm4 4790	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2178	. . 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 4609	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4783	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4799	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2185	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4609	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2178	. 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 1342   `'ccnv 4597   dom cdm 4598   ran crn 4599   o. ccom 4602

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609

This theorem is referenced by:  dfdm2  5132  foco  5414