Theorem 1stcof 6108

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 6102	. . . 4 |- 1st : _V -onto-> _V
2		fofn 5393	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 1st Fn _V
4		ffn 5318	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5320	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 121	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5343	. . 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 5091	. . 3 |- ran ( 1st o. F ) = ran ( 1st |` ran F )
10		frn 5327	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4892	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) )
12		rnss 4815	. . . . 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 6104	. . . . 5 |- ( 1st |` ( B X. C ) ) : ( B X. C ) --> B
15		frn 5327	. . . . 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 3140	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ B )
18	9, 17	eqsstrid 3174	. 2 |- ( F : A --> ( B X. C ) -> ran ( 1st o. F ) C_ B )
19		df-f 5173	. 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 2712   C_ wss 3102   X. cxp 4583   ran crn 4586   |` cres 4587   o. ccom 4589   Fn wfn 5164   -->wf 5165   -onto->wfo 5167   1stc1st 6083

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fo 5175  df-fv 5177  df-1st 6085

This theorem is referenced by: (None)