Theorem rncoeq 4884

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 4883	. 2 |- ( dom `' B = ran `' A -> dom ( `' B o. `' A ) = dom `' A )
2		eqcom 2172	. . 3 |- ( dom A = ran B <-> ran B = dom A )
3		df-rn 4622	. . . 4 |- ran B = dom `' B
4		dfdm4 4803	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 2184	. . 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 4622	. . . 4 |- ran ( A o. B ) = dom `' ( A o. B )
8		cnvco 4796	. . . . 5 |- `' ( A o. B ) = ( `' B o. `' A )
9	8	dmeqi 4812	. . . 4 |- dom `' ( A o. B ) = dom ( `' B o. `' A )
10	7, 9	eqtri 2191	. . 3 |- ran ( A o. B ) = dom ( `' B o. `' A )
11		df-rn 4622	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 2184	. 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 1348   `'ccnv 4610   dom cdm 4611   ran crn 4612   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622

This theorem is referenced by:  dfdm2  5145  foco  5430