Theorem coexg 2671

Description: The composition of two sets is a set.

Assertion

Ref	Expression
coexg	|- ((A e. C /\ B e. D) -> (A o. B) e. V)

Proof of Theorem coexg

Step	Hyp	Ref	Expression
1		relco 2658	. . 3 |- Rel (A o. B)
2		relssdr 2668	. . . 4 |- (Rel (A o. B) -> (A o. B) (_ (dom (A o. B) X. ran (A o. B)))
3		dmco 2570	. . . . . 6 |- dom (A o. B) (_ dom B
4		rnco 2571	. . . . . 6 |- ran (A o. B) (_ ran A
5		ssxp 2487	. . . . . 6 |- ((dom (A o. B) (_ dom B /\ ran (A o. B) (_ ran A) -> (dom (A o. B) X. ran (A o. B)) (_ (dom B X. ran A))
6	3, 4, 5	mp2an 520	. . . . 5 |- (dom (A o. B) X. ran (A o. B)) (_ (dom B X. ran A)
7		sstr2 1510	. . . . 5 |- ((A o. B) (_ (dom (A o. B) X. ran (A o. B)) -> ((dom (A o. B) X. ran (A o. B)) (_ (dom B X. ran A) -> (A o. B) (_ (dom B X. ran A)))
8	6, 7	mpi 44	. . . 4 |- ((A o. B) (_ (dom (A o. B) X. ran (A o. B)) -> (A o. B) (_ (dom B X. ran A))
9	2, 8	syl 12	. . 3 |- (Rel (A o. B) -> (A o. B) (_ (dom B X. ran A))
10	1, 9	ax-mp 6	. 2 |- (A o. B) (_ (dom B X. ran A)
11		dmexg 2551	. . . . 5 |- (B e. D -> dom B e. V)
12		rnexg 2569	. . . . 5 |- (A e. C -> ran A e. V)
13	11, 12	anim12i 268	. . . 4 |- ((B e. D /\ A e. C) -> (dom B e. V /\ ran A e. V))
14	13	ancoms 334	. . 3 |- ((A e. C /\ B e. D) -> (dom B e. V /\ ran A e. V))
15		xpexg 2489	. . 3 |- ((dom B e. V /\ ran A e. V) -> (dom B X. ran A) e. V)
16		ssexg 1702	. . 3 |- ((dom B X. ran A) e. V -> ((A o. B) (_ (dom B X. ran A) -> (A o. B) e. V))
17	14, 15, 16	3syl 21	. 2 |- ((A e. C /\ B e. D) -> ((A o. B) (_ (dom B X. ran A) -> (A o. B) e. V))
18	10, 17	mpi 44	1 |- ((A e. C /\ B e. D) -> (A o. B) e. V)

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  Vcvv 1348   (_ wss 1487   X. cxp 2408  dom cdm 2410  ran crn 2411   o. ccom 2414  Rel wrel 2415

This theorem is referenced by:  coex 2672

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429

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