Theorem cofunexg 3580

Description: Existence of a composition when the first member is a function.

Assertion

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

Proof of Theorem cofunexg

Step	Hyp	Ref	Expression
1		xpexg 3259	. . 3 |- ((dom ( A o. B) e. V /\ ran ( A o. B) e. V) -> (dom ( A o. B) X. ran ( A o. B)) e. V)
2		dmexg 3358	. . . . 5 |- (B e. C -> dom B e. V)
3		dmcoss 3363	. . . . . 6 |- dom ( A o. B) (_ dom B
4		ssexg 2721	. . . . . 6 |- ((dom ( A o. B) (_ dom B /\ dom B e. V) -> dom ( A o. B) e. V)
5	3, 4	mpan 695	. . . . 5 |- (dom B e. V -> dom ( A o. B) e. V)
6	2, 5	syl 10	. . . 4 |- (B e. C -> dom ( A o. B) e. V)
7	6	adantl 388	. . 3 |- ((Fun A /\ B e. C) -> dom ( A o. B) e. V)
8		resfunexg 3579	. . . . . 6 |- ((Fun A /\ ran B e. V) -> (A |` ran B) e. V)
9		rnexg 3359	. . . . . 6 |- (B e. C -> ran B e. V)
10	8, 9	sylan2 451	. . . . 5 |- ((Fun A /\ B e. C) -> (A |` ran B) e. V)
11		rnexg 3359	. . . . 5 |- ((A |` ran B) e. V -> ran ( A |` ran B) e. V)
12	10, 11	syl 10	. . . 4 |- ((Fun A /\ B e. C) -> ran ( A |` ran B) e. V)
13		rnco 3502	. . . 4 |- ran ( A o. B) = ran ( A |` ran B)
14	12, 13	syl5eqel 1552	. . 3 |- ((Fun A /\ B e. C) -> ran ( A o. B) e. V)
15	1, 7, 14	sylanc 471	. 2 |- ((Fun A /\ B e. C) -> (dom ( A o. B) X. ran ( A o. B)) e. V)
16		relco 3484	. . . 4 |- Rel (A o. B)
17		relssdr 3513	. . . 4 |- (Rel (A o. B) -> (A o. B) (_ (dom ( A o. B) X. ran ( A o. B)))
18	16, 17	ax-mp 7	. . 3 |- (A o. B) (_ (dom ( A o. B) X. ran ( A o. B))
19		ssexg 2721	. . 3 |- (((A o. B) (_ (dom ( A o. B) X. ran ( A o. B)) /\ (dom ( A o. B) X. ran ( A o. B)) e. V) -> (A o. B) e. V)
20	18, 19	mpan 695	. 2 |- ((dom ( A o. B) X. ran ( A o. B)) e. V -> (A o. B) e. V)
21	15, 20	syl 10	1 |- ((Fun A /\ B e. C) -> (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   |` cres 3172   o. ccom 3174  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  cofunex2g 3581

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-rep 2693  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-rex 1650  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-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192

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