Theorem cofunexg 3572

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 3254	. . 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 3352	. . . . 5 |- (B e. C -> dom B e. V)
3		dmcoss 3355	. . . . . 6 |- dom ( A o. B) (_ dom B
4		ssexg 2716	. . . . . 6 |- ((dom ( A o. B) (_ dom B /\ dom B e. V) -> dom ( A o. B) e. V)
5	3, 4	mpan 694	. . . . 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 3571	. . . . . 6 |- ((Fun A /\ ran B e. V) -> (A |` ran B) e. V)
9		rnexg 3353	. . . . . 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 3353	. . . . 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 3494	. . . 4 |- ran ( A o. B) = ran ( A |` ran B)
14	12, 13	syl5eqel 1549	. . 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 3476	. . . 4 |- Rel (A o. B)
17		relssdr 3505	. . . 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 2716	. . 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 694	. 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 956  Vcvv 1807   (_ wss 2043   X. cxp 3163  dom cdm 3165  ran crn 3166   |` cres 3167   o. ccom 3169  Rel wrel 3170  Fun wfun 3171

This theorem is referenced by:  cofunex2g 3573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187

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