Theorem 1stcof 6131

Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Assertion

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

Proof of Theorem 1stcof

Step	Hyp	Ref	Expression
1		fo1st 6125	. . . 4 |- 1st : _V -onto-> _V
2		fofn 5412	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 1st Fn _V
4		ffn 5337	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5339	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 121	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5362	. . 3 |- ( ( 1st Fn _V /\ F : A --> _V ) -> ( 1st o. F ) Fn A )
8	3, 6, 7	sylancr 411	. 2 |- ( F : A --> ( B X. C ) -> ( 1st o. F ) Fn A )
9		rnco 5110	. . 3 |- ran ( 1st o. F ) = ran ( 1st |` ran F )
10		frn 5346	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4911	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) )
12		rnss 4834	. . . . 5 |- ( ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) -> ran ( 1st |` ran F ) C_ ran ( 1st |` ( B X. C ) ) )
13	10, 11, 12	3syl 17	. . . 4 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ ran ( 1st |` ( B X. C ) ) )
14		f1stres 6127	. . . . 5 |- ( 1st |` ( B X. C ) ) : ( B X. C ) --> B
15		frn 5346	. . . . 5 |- ( ( 1st |` ( B X. C ) ) : ( B X. C ) --> B -> ran ( 1st |` ( B X. C ) ) C_ B )
16	14, 15	ax-mp 5	. . . 4 |- ran ( 1st |` ( B X. C ) ) C_ B
17	13, 16	sstrdi 3154	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ B )
18	9, 17	eqsstrid 3188	. 2 |- ( F : A --> ( B X. C ) -> ran ( 1st o. F ) C_ B )
19		df-f 5192	. 2 |- ( ( 1st o. F ) : A --> B <-> ( ( 1st o. F ) Fn A /\ ran ( 1st o. F ) C_ B ) )
20	8, 18, 19	sylanbrc 414	1 |- ( F : A --> ( B X. C ) -> ( 1st o. F ) : A --> B )

Syntax hints:   -> wi 4   _Vcvv 2726   C_ wss 3116   X. cxp 4602   ran crn 4605   |` cres 4606   o. ccom 4608   Fn wfn 5183   -->wf 5184   -onto->wfo 5186   1stc1st 6106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108

This theorem is referenced by: (None)