Theorem 2ndcof 6070

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 6064	. . . 4 |- 2nd : _V -onto-> _V
2		fofn 5355	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 2nd Fn _V
4		ffn 5280	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5282	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 121	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5305	. . 3 |- ( ( 2nd Fn _V /\ F : A --> _V ) -> ( 2nd o. F ) Fn A )
8	3, 6, 7	sylancr 411	. 2 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) Fn A )
9		rnco 5053	. . 3 |- ran ( 2nd o. F ) = ran ( 2nd |` ran F )
10		frn 5289	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4854	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) )
12		rnss 4777	. . . . 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 6066	. . . . 5 |- ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C
15		frn 5289	. . . . 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 3114	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ C )
18	9, 17	eqsstrid 3148	. 2 |- ( F : A --> ( B X. C ) -> ran ( 2nd o. F ) C_ C )
19		df-f 5135	. 2 |- ( ( 2nd o. F ) : A --> C <-> ( ( 2nd o. F ) Fn A /\ ran ( 2nd o. F ) C_ C ) )
20	8, 18, 19	sylanbrc 414	1 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )

Syntax hints:   -> wi 4   _Vcvv 2689   C_ wss 3076   X. cxp 4545   ran crn 4548   |` cres 4549   o. ccom 4551   Fn wfn 5126   -->wf 5127   -onto->wfo 5129   2ndc2nd 6045

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-2nd 6047

This theorem is referenced by: (None)