Theorem dmco 5210

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 4889	. 2 |- dom ( A o. B ) = ran `' ( A o. B )
2		cnvco 4881	. . 3 |- `' ( A o. B ) = ( `' B o. `' A )
3	2	rneqi 4925	. 2 |- ran `' ( A o. B ) = ran ( `' B o. `' A )
4		rnco2 5209	. . 3 |- ran ( `' B o. `' A ) = ( `' B " ran `' A )
5		dfdm4 4889	. . . 4 |- dom A = ran `' A
6	5	imaeq2i 5039	. . 3 |- ( `' B " dom A ) = ( `' B " ran `' A )
7	4, 6	eqtr4i 2231	. 2 |- ran ( `' B o. `' A ) = ( `' B " dom A )
8	1, 3, 7	3eqtri 2232	1 |- dom ( A o. B ) = ( `' B " dom A )

Syntax hints:   = wceq 1373   `'ccnv 4692   dom cdm 4693   ran crn 4694   "cima 4696   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  casedm  7214  caseinl  7219  caseinr  7220  djudm  7233