Theorem rncoeq 5031

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 5030	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2234	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4760	. . . 4 |- ran B = dom `' B
4		dfdm4 4948	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2246	. . 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 4760	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4940	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4957	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2253	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4760	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2246	. 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 4748   dom cdm 4749   ran crn 4750   o. ccom 4753

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760

This theorem is referenced by:  dfdm2  5297  foco  5601