Theorem 1stcof 6272

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 6266	. . . 4 |- 1st : _V -onto-> _V
2		fofn 5522	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 1st Fn _V
4		ffn 5445	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5447	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 122	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5472	. . 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 5208	. . 3 |- ran ( 1st o. F ) = ran ( 1st |` ran F )
10		frn 5454	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 5005	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 1st |` ran F ) C_ ( 1st |` ( B X. C ) ) )
12		rnss 4927	. . . . 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 6268	. . . . 5 |- ( 1st |` ( B X. C ) ) : ( B X. C ) --> B
15		frn 5454	. . . . 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 3213	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 1st |` ran F ) C_ B )
18	9, 17	eqsstrid 3247	. 2 |- ( F : A --> ( B X. C ) -> ran ( 1st o. F ) C_ B )
19		df-f 5294	. 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 2776   C_ wss 3174   X. cxp 4691   ran crn 4694   |` cres 4695   o. ccom 4697   Fn wfn 5285   -->wf 5286   -onto->wfo 5288   1stc1st 6247

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-1st 6249

This theorem is referenced by: (None)