Theorem coexg 4429

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		ssexg 3457	. 2 |- (((A o. B) C_ (dom B X. ran A) /\ (dom B X. ran A) e. _V) -> (A o. B) e. _V)
2		relco 4392	. . 3 |- Rel (A o. B)
3		relssdmrn 4416	. . . 4 |- (Rel (A o. B) -> (A o. B) C_ (dom ( A o. B) X. ran ( A o. B)))
4		dmcoss 4211	. . . . . 6 |- dom ( A o. B) C_ dom B
5		rncoss 4213	. . . . . 6 |- ran ( A o. B) C_ ran A
6		xpss12 4089	. . . . . 6 |- ((dom ( A o. B) C_ dom B /\ ran ( A o. B) C_ ran A) -> (dom ( A o. B) X. ran ( A o. B)) C_ (dom B X. ran A))
7	4, 5, 6	mp2an 761	. . . . 5 |- (dom ( A o. B) X. ran ( A o. B)) C_ (dom B X. ran A)
8		sstr2 2623	. . . . 5 |- ((A o. B) C_ (dom ( A o. B) X. ran ( A o. B)) -> ((dom ( A o. B) X. ran ( A o. B)) C_ (dom B X. ran A) -> (A o. B) C_ (dom B X. ran A)))
9	7, 8	mpi 55	. . . 4 |- ((A o. B) C_ (dom ( A o. B) X. ran ( A o. B)) -> (A o. B) C_ (dom B X. ran A))
10	3, 9	syl 12	. . 3 |- (Rel (A o. B) -> (A o. B) C_ (dom B X. ran A))
11	2, 10	ax-mp 7	. 2 |- (A o. B) C_ (dom B X. ran A)
12		xpexg 4095	. . . 4 |- ((dom B e. _V /\ ran A e. _V) -> (dom B X. ran A) e. _V)
13		dmexg 4206	. . . 4 |- (B e. D -> dom B e. _V)
14		rnexg 4207	. . . 4 |- (A e. C -> ran A e. _V)
15	12, 13, 14	syl2an 503	. . 3 |- ((B e. D /\ A e. C) -> (dom B X. ran A) e. _V)
16	15	ancoms 484	. 2 |- ((A e. C /\ B e. D) -> (dom B X. ran A) e. _V)
17	1, 11, 16	sylancr 526	1 |- ((A e. C /\ B e. D) -> (A o. B) e. _V)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  _Vcvv 2292   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   o. ccom 3990  Rel wrel 3991

This theorem is referenced by:  coex 4430  fodomfi 5656  symgoprv 10203  mapmapmap 14486  injsurinj 14487  cmphmp 14878  hmphtr 14885  hmeogrp 14892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005

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