Theorem coexg 3524

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 3484	. . 3 |- Rel (A o. B)
2		relssdr 3513	. . . 4 |- (Rel (A o. B) -> (A o. B) (_ (dom ( A o. B) X. ran ( A o. B)))
3		dmcoss 3363	. . . . . 6 |- dom ( A o. B) (_ dom B
4		rncoss 3364	. . . . . 6 |- ran ( A o. B) (_ ran A
5		ssxp 3256	. . . . . 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 697	. . . . 5 |- (dom ( A o. B) X. ran ( A o. B)) (_ (dom B X. ran A)
7		sstr2 2071	. . . . 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 2721	. . 3 |- (((A o. B) (_ (dom B X. ran A) /\ (dom B X. ran A) e. V) -> (A o. B) e. V)
12		xpexg 3259	. . . . 5 |- ((dom B e. V /\ ran A e. V) -> (dom B X. ran A) e. V)
13		dmexg 3358	. . . . 5 |- (B e. D -> dom B e. V)
14		rnexg 3359	. . . . 5 |- (A e. C -> ran A e. V)
15	12, 13, 14	syl2an 454	. . . 4 |- ((B e. D /\ A e. C) -> (dom B X. ran A) e. V)
16	15	ancoms 436	. . 3 |- ((A e. C /\ B e. D) -> (dom B X. ran A) e. V)
17	11, 16	sylan2 451	. 2 |- (((A o. B) (_ (dom B X. ran A) /\ (A e. C /\ B e. D)) -> (A o. B) e. V)
18	10, 17	mpan 695	1 |- ((A e. C /\ B e. D) -> (A o. B) e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171   o. ccom 3174  Rel wrel 3175

This theorem is referenced by:  coex 3525  fodomfiOLD 4566  symgoprval 10404  cmphmp 10521  hmphtr 10531  hmeogrp 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189

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