Theorem 2ndcof 6371

Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
2ndcof	|- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )

Proof of Theorem 2ndcof

Step	Hyp	Ref	Expression
1		fo2nd 6365	. . . 4 |- 2nd : _V -onto-> _V
2		fofn 5597	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 2nd Fn _V
4		ffn 5513	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5515	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 122	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5544	. . 3 |- ( ( 2nd Fn _V /\ F : A --> _V ) -> ( 2nd o. F ) Fn A )
8	3, 6, 7	sylancr 414	. 2 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) Fn A )
9		rnco 5274	. . 3 |- ran ( 2nd o. F ) = ran ( 2nd |` ran F )
10		frn 5522	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 5070	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) )
12		rnss 4992	. . . . 5 |- ( ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) -> ran ( 2nd |` ran F ) C_ ran ( 2nd |` ( B X. C ) ) )
13	10, 11, 12	3syl 17	. . . 4 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ ran ( 2nd |` ( B X. C ) ) )
14		f2ndres 6367	. . . . 5 |- ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C
15		frn 5522	. . . . 5 |- ( ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C -> ran ( 2nd |` ( B X. C ) ) C_ C )
16	14, 15	ax-mp 5	. . . 4 |- ran ( 2nd |` ( B X. C ) ) C_ C
17	13, 16	sstrdi 3254	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ C )
18	9, 17	eqsstrid 3288	. 2 |- ( F : A --> ( B X. C ) -> ran ( 2nd o. F ) C_ C )
19		df-f 5361	. 2 |- ( ( 2nd o. F ) : A --> C <-> ( ( 2nd o. F ) Fn A /\ ran ( 2nd o. F ) C_ C ) )
20	8, 18, 19	sylanbrc 417	1 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )

Syntax hints:   -> wi 4   _Vcvv 2815   C_ wss 3214   X. cxp 4752   ran crn 4755   |` cres 4756   o. ccom 4758   Fn wfn 5352   -->wf 5353   -onto->wfo 5355   2ndc2nd 6346

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

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 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-2nd 6348

This theorem is referenced by: (None)