Theorem 1stcof 5040

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		df-f 4010	. 2 |- ((1st o. F):A-->B <-> ((1st o. F) Fn A /\ ran (1st o. F) C_ B))
2		fnfco 4581	. . 3 |- ((1st Fn _V /\ F:A-->_V) -> (1st o. F) Fn A)
3		fo1st 5032	. . . 4 |- 1st:_V-onto->_V
4		fofn 4619	. . . 4 |- (1st:_V-onto->_V -> 1st Fn _V)
5	3, 4	ax-mp 7	. . 3 |- 1st Fn _V
6		ffn 4562	. . . 4 |- (F:A-->(B X. C) -> F Fn A)
7		dffn2 4563	. . . 4 |- (F Fn A <-> F:A-->_V)
8	6, 7	sylib 215	. . 3 |- (F:A-->(B X. C) -> F:A-->_V)
9	2, 5, 8	sylancr 526	. 2 |- (F:A-->(B X. C) -> (1st o. F) Fn A)
10		frn 4569	. . . . 5 |- (F:A-->(B X. C) -> ran F C_ (B X. C))
11		ssres2 4240	. . . . 5 |- (ran F C_ (B X. C) -> (1st |` ran F) C_ (1st |` (B X. C)))
12		rnss 4189	. . . . 5 |- ((1st |` ran F) C_ (1st |` (B X. C)) -> ran (1st |` ran F) C_ ran (1st |` (B X. C)))
13	10, 11, 12	3syl 24	. . . 4 |- (F:A-->(B X. C) -> ran (1st |` ran F) C_ ran (1st |` (B X. C)))
14		f1stres 5034	. . . . . 6 |- (1st |` (B X. C)):(B X. C)-->B
15		frn 4569	. . . . . 6 |- ((1st |` (B X. C)):(B X. C)-->B -> ran (1st |` (B X. C)) C_ B)
16	14, 15	ax-mp 7	. . . . 5 |- ran (1st |` (B X. C)) C_ B
17	16	a1i 8	. . . 4 |- (F:A-->(B X. C) -> ran (1st |` (B X. C)) C_ B)
18	13, 17	sstrd 2627	. . 3 |- (F:A-->(B X. C) -> ran (1st |` ran F) C_ B)
19		rnco 4404	. . 3 |- ran (1st o. F) = ran (1st |` ran F)
20	18, 19	syl5ss 2661	. 2 |- (F:A-->(B X. C) -> ran (1st o. F) C_ B)
21	1, 9, 20	sylanbrc 527	1 |- (F:A-->(B X. C) -> (1st o. F):A-->B)

Syntax hints:   -> wi 3  _Vcvv 2292   C_ wss 2593   X. cxp 3984  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  1stc1st 5018

This theorem is referenced by:  bcthlem22 9298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-1st 5020

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