Theorem 2ndcof 7197

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 7189	. . . 4 |- 2nd : _V -onto-> _V
2		fofn 6117	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3	1, 2	ax-mp 5	. . 3 |- 2nd Fn _V
4		ffn 6045	. . . 4 |- ( F : A --> ( B X. C ) -> F Fn A )
5		dffn2 6047	. . . 4 |- ( F Fn A <-> F : A --> _V )
6	4, 5	sylib 208	. . 3 |- ( F : A --> ( B X. C ) -> F : A --> _V )
7		fnfco 6069	. . 3 |- ( ( 2nd Fn _V /\ F : A --> _V ) -> ( 2nd o. F ) Fn A )
8	3, 6, 7	sylancr 695	. 2 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) Fn A )
9		rnco 5641	. . 3 |- ran ( 2nd o. F ) = ran ( 2nd |` ran F )
10		frn 6053	. . . . 5 |- ( F : A --> ( B X. C ) -> ran F C_ ( B X. C ) )
11		ssres2 5425	. . . . 5 |- ( ran F C_ ( B X. C ) -> ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) )
12		rnss 5354	. . . . 5 |- ( ( 2nd |` ran F ) C_ ( 2nd |` ( B X. C ) ) -> ran ( 2nd |` ran F ) C_ ran ( 2nd |` ( B X. C ) ) )
13	10, 11, 12	3syl 18	. . . 4 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ ran ( 2nd |` ( B X. C ) ) )
14		f2ndres 7191	. . . . 5 |- ( 2nd |` ( B X. C ) ) : ( B X. C ) --> C
15		frn 6053	. . . . 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	syl6ss 3615	. . 3 |- ( F : A --> ( B X. C ) -> ran ( 2nd |` ran F ) C_ C )
18	9, 17	syl5eqss 3649	. 2 |- ( F : A --> ( B X. C ) -> ran ( 2nd o. F ) C_ C )
19		df-f 5892	. 2 |- ( ( 2nd o. F ) : A --> C <-> ( ( 2nd o. F ) Fn A /\ ran ( 2nd o. F ) C_ C ) )
20	8, 18, 19	sylanbrc 698	1 |- ( F : A --> ( B X. C ) -> ( 2nd o. F ) : A --> C )

Syntax hints:   -> wi 4   _Vcvv 3200   C_ wss 3574   X. cxp 5112   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-2nd 7169

This theorem is referenced by:  axdc4lem  9277