Theorem rncoeq 4939

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 4938	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2198	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4674	. . . 4 |- ran B = dom `' B
4		dfdm4 4858	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2210	. . 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 4674	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4851	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4867	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2217	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4674	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2210	. 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 1364   `'ccnv 4662   dom cdm 4663   ran crn 4664   o. ccom 4667

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674

This theorem is referenced by:  dfdm2  5204  foco  5491