Theorem cofunexg 3576

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 3255	. . 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 3354	. . . . 5 |- (B e. C -> dom B e. V)
3		dmcoss 3359	. . . . . 6 |- dom ( A o. B) (_ dom B
4		ssexg 2717	. . . . . 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 3575	. . . . . 6 |- ((Fun A /\ ran B e. V) -> (A |` ran B) e. V)
9		rnexg 3355	. . . . . 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 3355	. . . . 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 3498	. . . 4 |- ran ( A o. B) = ran ( A |` ran B)
14	12, 13	syl5eqel 1550	. . 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 3480	. . . 4 |- Rel (A o. B)
17		relssdr 3509	. . . 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 2717	. . 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 957  Vcvv 1808   (_ wss 2044   X. cxp 3164  dom cdm 3166  ran crn 3167   |` cres 3168   o. ccom 3170  Rel wrel 3171  Fun wfun 3172

This theorem is referenced by:  cofunex2g 3577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188

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