Theorem 2ndcof 6250

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 6244	. . . 4 |- 2nd : _V -onto-> _V
2		fofn 5500	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 2nd Fn _V
4		ffn 5425	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 5427	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 122	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 5450	. . 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 5189	. . 3 |- ran ( 2nd o. F ) = ran ( 2nd |` ran F )
10		frn 5434	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 4986	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) )
12		rnss 4908	. . . . 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 6246	. . . . 5 |- ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C
15		frn 5434	. . . . 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 3205	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ C )
18	9, 17	eqsstrid 3239	. 2 |- ( F : A --> ( B X. C ) -> ran ( 2nd o. F ) C_ C )
19		df-f 5275	. 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 2772   C_ wss 3166   X. cxp 4673   ran crn 4676   |` cres 4677   o. ccom 4679   Fn wfn 5266   -->wf 5267   -onto->wfo 5269   2ndc2nd 6225

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-2nd 6227

This theorem is referenced by: (None)