Theorem 1stcof 6262

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 6256	. . . 4 |- 1st : _V -onto-> _V
2		fofn 5512	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 1st Fn _V
4		ffn 5435	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5437	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 122	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5462	. . 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 5198	. . 3 |- ran ( 1st o. F ) = ran ( 1st |` ran F )
10		frn 5444	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4995	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) )
12		rnss 4917	. . . . 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 6258	. . . . 5 |- ( 1st |` ( B X. C ) ) : ( B X. C ) --> B
15		frn 5444	. . . . 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 3209	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ B )
18	9, 17	eqsstrid 3243	. 2 |- ( F : A --> ( B X. C ) -> ran ( 1st o. F ) C_ B )
19		df-f 5284	. 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 2773   C_ wss 3170   X. cxp 4681   ran crn 4684   |` cres 4685   o. ccom 4687   Fn wfn 5275   -->wf 5276   -onto->wfo 5278   1stc1st 6237

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-1st 6239

This theorem is referenced by: (None)