Theorem ply1frcl 19683

Description: Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)

Hypothesis

Ref	Expression
ply1frcl.q	|- Q = ran ( S evalSub1 R )

Assertion

Ref	Expression
ply1frcl	|- ( X e. Q -> ( S e. _V /\ R e. ~P ( Base ` S ) ) )

Proof of Theorem ply1frcl

Dummy variables r b s x y e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ne0i 3921	. . 3 |- ( X e. ran ( S evalSub1 R ) -> ran ( S evalSub1 R ) =/= (/) )
2		ply1frcl.q	. . 3 |- Q = ran ( S evalSub1 R )
3	1, 2	eleq2s 2719	. 2 |- ( X e. Q -> ran ( S evalSub1 R ) =/= (/) )
4		rneq 5351	. . . 4 |- ( ( S evalSub1 R ) = (/) -> ran ( S evalSub1 R ) = ran (/) )
5		rn0 5377	. . . 4 |- ran (/) = (/)
6	4, 5	syl6eq 2672	. . 3 |- ( ( S evalSub1 R ) = (/) -> ran ( S evalSub1 R ) = (/) )
7	6	necon3i 2826	. 2 |- ( ran ( S evalSub1 R ) =/= (/) -> ( S evalSub1 R ) =/= (/) )
8		n0 3931	. . 3 |- ( ( S evalSub1 R ) =/= (/) <-> E. e e e. ( S evalSub1 R ) )
9		df-evls1 19680	. . . . . . 7 |- evalSub1 = ( s e. _V , r e. ~P ( Base ` s ) |-> [_ ( Base ` s ) / b ]_ ( ( x e. ( b ^m ( b ^m 1o ) ) |-> ( x o. ( y e. b |-> ( 1o X. { y } ) ) ) ) o. ( ( 1o evalSub s ) ` r ) ) )
10	9	dmmpt2ssx 7235	. . . . . 6 |- dom evalSub1 C_ U_ s e. _V ( { s } X. ~P ( Base ` s ) )
11		elfvdm 6220	. . . . . . 7 |- ( e e. ( evalSub1 ` <. S , R >. ) -> <. S , R >. e. dom evalSub1 )
12		df-ov 6653	. . . . . . 7 |- ( S evalSub1 R ) = ( evalSub1 ` <. S , R >. )
13	11, 12	eleq2s 2719	. . . . . 6 |- ( e e. ( S evalSub1 R ) -> <. S , R >. e. dom evalSub1 )
14	10, 13	sseldi 3601	. . . . 5 |- ( e e. ( S evalSub1 R ) -> <. S , R >. e. U_ s e. _V ( { s } X. ~P ( Base ` s ) ) )
15		fveq2 6191	. . . . . . 7 |- ( s = S -> ( Base ` s ) = ( Base ` S ) )
16	15	pweqd 4163	. . . . . 6 |- ( s = S -> ~P ( Base ` s ) = ~P ( Base ` S ) )
17	16	opeliunxp2 5260	. . . . 5 |- ( <. S , R >. e. U_ s e. _V ( { s } X. ~P ( Base ` s ) ) <-> ( S e. _V /\ R e. ~P ( Base ` S ) ) )
18	14, 17	sylib 208	. . . 4 |- ( e e. ( S evalSub1 R ) -> ( S e. _V /\ R e. ~P ( Base ` S ) ) )
19	18	exlimiv 1858	. . 3 |- ( E. e e e. ( S evalSub1 R ) -> ( S e. _V /\ R e. ~P ( Base ` S ) ) )
20	8, 19	sylbi 207	. 2 |- ( ( S evalSub1 R ) =/= (/) -> ( S e. _V /\ R e. ~P ( Base ` S ) ) )
21	3, 7, 20	3syl 18	1 |- ( X e. Q -> ( S e. _V /\ R e. ~P ( Base ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   [_csb 3533   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   evalSub ces 19504   evalSub₁ ces1 19678

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-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-pw 4160  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-evls1 19680

This theorem is referenced by: (None)