Theorem dmco 5119

Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Assertion

Ref	Expression
dmco	|- dom ( A o. B ) = ( `' B " dom A )

Proof of Theorem dmco

Step	Hyp	Ref	Expression
1		dfdm4 4803	. 2 |- dom ( A o. B ) = ran `' ( A o. B )
2		cnvco 4796	. . 3 |- `' ( A o. B ) = ( `' B o. `' A )
3	2	rneqi 4839	. 2 |- ran `' ( A o. B ) = ran ( `' B o. `' A )
4		rnco2 5118	. . 3 |- ran ( `' B o. `' A ) = ( `' B " ran `' A )
5		dfdm4 4803	. . . 4 |- dom A = ran `' A
6	5	imaeq2i 4951	. . 3 |- ( `' B " dom A ) = ( `' B " ran `' A )
7	4, 6	eqtr4i 2194	. 2 |- ran ( `' B o. `' A ) = ( `' B " dom A )
8	1, 3, 7	3eqtri 2195	1 |- dom ( A o. B ) = ( `' B " dom A )

Syntax hints:   = wceq 1348   `'ccnv 4610   dom cdm 4611   ran crn 4612   "cima 4614   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-ral 2453  df-rex 2454  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-xp 4617  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  casedm  7063  caseinl  7068  caseinr  7069  djudm  7082