Theorem coexg 3510

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 3470	. . 3 |- Rel (A o. B)
2		relssdr 3499	. . . 4 |- (Rel (A o. B) -> (A o. B) (_ (dom ( A o. B) X. ran ( A o. B)))
3		dmcoss 3347	. . . . . 6 |- dom ( A o. B) (_ dom B
4		rncoss 3348	. . . . . 6 |- ran ( A o. B) (_ ran A
5		ssxp 3246	. . . . . 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 695	. . . . 5 |- (dom ( A o. B) X. ran ( A o. B)) (_ (dom B X. ran A)
7		sstr2 2061	. . . . 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 2711	. . 3 |- (((A o. B) (_ (dom B X. ran A) /\ (dom B X. ran A) e. V) -> (A o. B) e. V)
12		xpexg 3249	. . . . 5 |- ((dom B e. V /\ ran A e. V) -> (dom B X. ran A) e. V)
13		dmexg 3344	. . . . 5 |- (B e. D -> dom B e. V)
14		rnexg 3345	. . . . 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 693	1 |- ((A e. C /\ B e. D) -> (A o. B) e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  Vcvv 1802   (_ wss 2037   X. cxp 3158  dom cdm 3160  ran crn 3161   o. ccom 3164  Rel wrel 3165

This theorem is referenced by:  coex 3511  fodomfi 4540  symgoprval 10309  cmphmp 10408  hmphtr 10418  hmeogrp 10425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179

Copyright terms: Public domain