Theorem 2ndcof 6055

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 6049	. . . 4 |- 2nd : _V -onto-> _V
2		fofn 5342	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 2nd Fn _V
4		ffn 5267	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5269	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 121	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5292	. . 3 |- ( ( 2nd Fn _V /\ F : A --> _V ) -> ( 2nd o. F ) Fn A )
8	3, 6, 7	sylancr 410	. 2 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) Fn A )
9		rnco 5040	. . 3 |- ran ( 2nd o. F ) = ran ( 2nd |` ran F )
10		frn 5276	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4841	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) )
12		rnss 4764	. . . . 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 6051	. . . . 5 |- ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C
15		frn 5276	. . . . 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 3104	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ C )
18	9, 17	eqsstrid 3138	. 2 |- ( F : A --> ( B X. C ) -> ran ( 2nd o. F ) C_ C )
19		df-f 5122	. 2 |- ( ( 2nd o. F ) : A --> C <-> ( ( 2nd o. F ) Fn A /\ ran ( 2nd o. F ) C_ C ) )
20	8, 18, 19	sylanbrc 413	1 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )

Syntax hints:   -> wi 4   _Vcvv 2681   C_ wss 3066   X. cxp 4532   ran crn 4535   |` cres 4536   o. ccom 4538   Fn wfn 5113   -->wf 5114   -onto->wfo 5116   2ndc2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-2nd 6032

This theorem is referenced by: (None)