Theorem rncoeq 4623

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 4622	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2083	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4374	. . . 4 |- ran B = dom `' B
4		dfdm4 4545	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2094	. . 3 |- ( ran B = dom A <-> dom `' B = ran `' A )
6	2, 5	bitri 182	. 2 |- ( dom A = ran B <-> dom `' B = ran `' A )
7		df-rn 4374	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4538	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4554	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2101	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4374	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2094	. 2 |- ( ran ( A o. B ) = ran A <-> dom ( `' B o. `' A ) = dom `' A )
13	1, 6, 12	3imtr4i 199	1 |- ( dom A = ran B -> ran ( A o. B ) = ran A )

Syntax hints:   -> wi 4   = wceq 1284   `'ccnv 4362   dom cdm 4363   ran crn 4364   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374

This theorem is referenced by:  dfdm2  4872  foco  5136