Theorem rncoeq 5051

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 5050	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2240	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4780	. . . 4 |- ran B = dom `' B
4		dfdm4 4968	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2252	. . 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 4780	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4960	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4977	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2259	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4780	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2252	. 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 1402   `'ccnv 4768   dom cdm 4769   ran crn 4770   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780

This theorem is referenced by:  dfdm2  5317  foco  5621