Theorem dmco 5245

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 4923	. 2 |- dom ( A o. B ) = ran `' ( A o. B )
2		cnvco 4915	. . 3 |- `' ( A o. B ) = ( `' B o. `' A )
3	2	rneqi 4960	. 2 |- ran `' ( A o. B ) = ran ( `' B o. `' A )
4		rnco2 5244	. . 3 |- ran ( `' B o. `' A ) = ( `' B " ran `' A )
5		dfdm4 4923	. . . 4 |- dom A = ran `' A
6	5	imaeq2i 5074	. . 3 |- ( `' B " dom A ) = ( `' B " ran `' A )
7	4, 6	eqtr4i 2255	. 2 |- ran ( `' B o. `' A ) = ( `' B " dom A )
8	1, 3, 7	3eqtri 2256	1 |- dom ( A o. B ) = ( `' B " dom A )

Syntax hints:   = wceq 1397   `'ccnv 4724   dom cdm 4725   ran crn 4726   "cima 4728   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  fncofn  5831  casedm  7284  caseinl  7289  caseinr  7290  djudm  7303