Theorem dmco 5271

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 4948	. 2 |- dom ( A o. B ) = ran `' ( A o. B )
2		cnvco 4940	. . 3 |- `' ( A o. B ) = ( `' B o. `' A )
3	2	rneqi 4985	. 2 |- ran `' ( A o. B ) = ran ( `' B o. `' A )
4		rnco2 5270	. . 3 |- ran ( `' B o. `' A ) = ( `' B " ran `' A )
5		dfdm4 4948	. . . 4 |- dom A = ran `' A
6	5	imaeq2i 5099	. . 3 |- ( `' B " dom A ) = ( `' B " ran `' A )
7	4, 6	eqtr4i 2256	. 2 |- ran ( `' B o. `' A ) = ( `' B " dom A )
8	1, 3, 7	3eqtri 2257	1 |- dom ( A o. B ) = ( `' B " dom A )

Syntax hints:   = wceq 1398   `'ccnv 4748   dom cdm 4749   ran crn 4750   "cima 4752   o. ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  fncofn  5862  casedm  7377  caseinl  7382  caseinr  7383  djudm  7396