Theorem s1co 13579

Description: Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Assertion

Ref	Expression
s1co	|- ( ( S e. A /\ F : A --> B ) -> ( F o. <" S "> ) = <" ( F ` S ) "> )

Proof of Theorem s1co

Step	Hyp	Ref	Expression
1		s1val 13378	. . . . 5 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		0cn 10032	. . . . . 6 |- 0 e. CC
3		xpsng 6406	. . . . . 6 |- ( ( 0 e. CC /\ S e. A ) -> ( { 0 } X. { S } ) = { <. 0 , S >. } )
4	2, 3	mpan 706	. . . . 5 |- ( S e. A -> ( { 0 } X. { S } ) = { <. 0 , S >. } )
5	1, 4	eqtr4d 2659	. . . 4 |- ( S e. A -> <" S "> = ( { 0 } X. { S } ) )
6	5	adantr 481	. . 3 |- ( ( S e. A /\ F : A --> B ) -> <" S "> = ( { 0 } X. { S } ) )
7	6	coeq2d 5284	. 2 |- ( ( S e. A /\ F : A --> B ) -> ( F o. <" S "> ) = ( F o. ( { 0 } X. { S } ) ) )
8		ffn 6045	. . . 4 |- ( F : A --> B -> F Fn A )
9		id 22	. . . 4 |- ( S e. A -> S e. A )
10		fcoconst 6401	. . . 4 |- ( ( F Fn A /\ S e. A ) -> ( F o. ( { 0 } X. { S } ) ) = ( { 0 } X. { ( F ` S ) } ) )
11	8, 9, 10	syl2anr 495	. . 3 |- ( ( S e. A /\ F : A --> B ) -> ( F o. ( { 0 } X. { S } ) ) = ( { 0 } X. { ( F ` S ) } ) )
12		fvex 6201	. . . . 5 |- ( F ` S ) e. _V
13		s1val 13378	. . . . 5 |- ( ( F ` S ) e. _V -> <" ( F ` S ) "> = { <. 0 , ( F ` S ) >. } )
14	12, 13	ax-mp 5	. . . 4 |- <" ( F ` S ) "> = { <. 0 , ( F ` S ) >. }
15		c0ex 10034	. . . . 5 |- 0 e. _V
16	15, 12	xpsn 6407	. . . 4 |- ( { 0 } X. { ( F ` S ) } ) = { <. 0 , ( F ` S ) >. }
17	14, 16	eqtr4i 2647	. . 3 |- <" ( F ` S ) "> = ( { 0 } X. { ( F ` S ) } )
18	11, 17	syl6reqr 2675	. 2 |- ( ( S e. A /\ F : A --> B ) -> <" ( F ` S ) "> = ( F o. ( { 0 } X. { S } ) ) )
19	7, 18	eqtr4d 2659	1 |- ( ( S e. A /\ F : A --> B ) -> ( F o. <" S "> ) = <" ( F ` S ) "> )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   X. cxp 5112   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888   CCcc 9934   0cc0 9936   <"cs1 13294

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-pow 4843  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

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-reu 2919  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-s1 13302

This theorem is referenced by:  cats1co  13601  s2co  13665  frmdgsum  17399  frmdup2  17402  efginvrel2  18140  vrgpinv  18182  frgpup2  18189  mrsubcv  31407