Theorem coexg 3530

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 3490	. . 3 |- Rel (A o. B)
2		relssdr 3519	. . . 4 |- (Rel (A o. B) -> (A o. B) (_ (dom ( A o. B) X. ran ( A o. B)))
3		dmcoss 3369	. . . . . 6 |- dom ( A o. B) (_ dom B
4		rncoss 3370	. . . . . 6 |- ran ( A o. B) (_ ran A
5		ssxp 3262	. . . . . 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 699	. . . . 5 |- (dom ( A o. B) X. ran ( A o. B)) (_ (dom B X. ran A)
7		sstr2 2074	. . . . 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 10	. . 3 |- (Rel (A o. B) -> (A o. B) (_ (dom B X. ran A))
10	1, 9	ax-mp 7	. 2 |- (A o. B) (_ (dom B X. ran A)
11		ssexg 2726	. . 3 |- (((A o. B) (_ (dom B X. ran A) /\ (dom B X. ran A) e. V) -> (A o. B) e. V)
12		xpexg 3265	. . . . 5 |- ((dom B e. V /\ ran A e. V) -> (dom B X. ran A) e. V)
13		dmexg 3364	. . . . 5 |- (B e. D -> dom B e. V)
14		rnexg 3365	. . . . 5 |- (A e. C -> ran A e. V)
15	12, 13, 14	syl2an 456	. . . 4 |- ((B e. D /\ A e. C) -> (dom B X. ran A) e. V)
16	15	ancoms 438	. . 3 |- ((A e. C /\ B e. D) -> (dom B X. ran A) e. V)
17	11, 16	sylan2 453	. 2 |- (((A o. B) (_ (dom B X. ran A) /\ (A e. C /\ B e. D)) -> (A o. B) e. V)
18	10, 17	mpan 697	1 |- ((A e. C /\ B e. D) -> (A o. B) e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  Vcvv 1814   (_ wss 2050   X. cxp 3174  dom cdm 3176  ran crn 3177   o. ccom 3180  Rel wrel 3181

This theorem is referenced by:  coex 3531  fodomfi 4575  fodomfiOLD 4576  symgoprval 10399  cmphmp 10507  hmphtr 10517  hmeogrp 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195

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