Theorem 1stcof 4091

Description: Composition of the first member function with another function.

Assertion

Ref	Expression
1stcof	|- (F:A-->(B X. C) -> (1st o. F):A-->B)

Proof of Theorem 1stcof

Step	Hyp	Ref	Expression
1		ffn 3619	. . . . 5 |- (F:A-->(B X. C) -> F Fn A)
2		fnf 3620	. . . . 5 |- (F Fn A <-> F:A-->V)
3	1, 2	sylib 198	. . . 4 |- (F:A-->(B X. C) -> F:A-->V)
4		fo1st 4081	. . . . . . 7 |- 1st:V-onto->V
5		fof 3663	. . . . . . 7 |- (1st:V-onto->V -> 1st:V-->V)
6	4, 5	ax-mp 7	. . . . . 6 |- 1st:V-->V
7		ffn 3619	. . . . . 6 |- (1st:V-->V -> 1st Fn V)
8	6, 7	ax-mp 7	. . . . 5 |- 1st Fn V
9		fnfco 3633	. . . . 5 |- ((1st Fn V /\ F:A-->V) -> (1st o. F) Fn A)
10	8, 9	mpan 694	. . . 4 |- (F:A-->V -> (1st o. F) Fn A)
11	3, 10	syl 10	. . 3 |- (F:A-->(B X. C) -> (1st o. F) Fn A)
12		frn 3624	. . . . . 6 |- (F:A-->(B X. C) -> ran F (_ (B X. C))
13		ssres2 3378	. . . . . 6 |- (ran F (_ (B X. C) -> (1st |` ran F) (_ (1st |` (B X. C)))
14		rnss 3337	. . . . . 6 |- ((1st |` ran F) (_ (1st |` (B X. C)) -> ran (1st |` ran F) (_ ran (1st |` (B X. C)))
15	12, 13, 14	3syl 20	. . . . 5 |- (F:A-->(B X. C) -> ran (1st |` ran F) (_ ran (1st |` (B X. C)))
16		f1stres 4083	. . . . . . 7 |- (1st |` (B X. C)):(B X. C)-->B
17		frn 3624	. . . . . . 7 |- ((1st |` (B X. C)):(B X. C)-->B -> ran (1st |` (B X. C)) (_ B)
18	16, 17	ax-mp 7	. . . . . 6 |- ran (1st |` (B X. C)) (_ B
19	18	a1i 8	. . . . 5 |- (F:A-->(B X. C) -> ran (1st |` (B X. C)) (_ B)
20	15, 19	sstrd 2070	. . . 4 |- (F:A-->(B X. C) -> ran (1st |` ran F) (_ B)
21		rnco 3494	. . . 4 |- ran (1st o. F) = ran (1st |` ran F)
22	20, 21	syl5ss 2101	. . 3 |- (F:A-->(B X. C) -> ran (1st o. F) (_ B)
23	11, 22	jca 288	. 2 |- (F:A-->(B X. C) -> ((1st o. F) Fn A /\ ran (1st o. F) (_ B))
24		df-f 3189	. 2 |- ((1st o. F):A-->B <-> ((1st o. F) Fn A /\ ran (1st o. F) (_ B))
25	23, 24	sylibr 200	1 |- (F:A-->(B X. C) -> (1st o. F):A-->B)

Syntax hints:   -> wi 3   /\ wa 223  Vcvv 1807   (_ wss 2043   X. cxp 3163  ran crn 3166   |` cres 3167   o. ccom 3169   Fn wfn 3172  -->wf 3173  -onto->wfo 3175  1stc1st 4067

This theorem is referenced by:  bcthlem22 7970

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-9 963  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-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  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-ral 1646  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  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-1st 4069

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