Theorem dmco 5237

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 4915	. 2 |- dom ( A o. B ) = ran `' ( A o. B )
2		cnvco 4907	. . 3 |- `' ( A o. B ) = ( `' B o. `' A )
3	2	rneqi 4952	. 2 |- ran `' ( A o. B ) = ran ( `' B o. `' A )
4		rnco2 5236	. . 3 |- ran ( `' B o. `' A ) = ( `' B " ran `' A )
5		dfdm4 4915	. . . 4 |- dom A = ran `' A
6	5	imaeq2i 5066	. . 3 |- ( `' B " dom A ) = ( `' B " ran `' A )
7	4, 6	eqtr4i 2253	. 2 |- ran ( `' B o. `' A ) = ( `' B " dom A )
8	1, 3, 7	3eqtri 2254	1 |- dom ( A o. B ) = ( `' B " dom A )

Syntax hints:   = wceq 1395   `'ccnv 4718   dom cdm 4719   ran crn 4720   "cima 4722   o. ccom 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  casedm  7253  caseinl  7258  caseinr  7259  djudm  7272