Theorem 1stcof 6359

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 6353	. . . 4 |- 1st : _V -onto-> _V
2		fofn 5594	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 1st Fn _V
4		ffn 5510	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5512	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 122	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5541	. . 3 |- ( ( 1st Fn _V /\ F : A --> _V ) -> ( 1st o. F ) Fn A )
8	3, 6, 7	sylancr 414	. 2 |- ( F : A --> ( B X. C ) -> ( 1st o. F ) Fn A )
9		rnco 5271	. . 3 |- ran ( 1st o. F ) = ran ( 1st |` ran F )
10		frn 5519	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 5067	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) )
12		rnss 4989	. . . . 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 6355	. . . . 5 |- ( 1st |` ( B X. C ) ) : ( B X. C ) --> B
15		frn 5519	. . . . 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 3252	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ B )
18	9, 17	eqsstrid 3286	. 2 |- ( F : A --> ( B X. C ) -> ran ( 1st o. F ) C_ B )
19		df-f 5358	. 2 |- ( ( 1st o. F ) : A --> B <-> ( ( 1st o. F ) Fn A /\ ran ( 1st o. F ) C_ B ) )
20	8, 18, 19	sylanbrc 417	1 |- ( F : A --> ( B X. C ) -> ( 1st o. F ) : A --> B )

Syntax hints:   -> wi 4   _Vcvv 2815   C_ wss 3213   X. cxp 4749   ran crn 4752   |` cres 4753   o. ccom 4755   Fn wfn 5349   -->wf 5350   -onto->wfo 5352   1stc1st 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-1st 6336

This theorem is referenced by: (None)