Theorem uffix 21725

Description: Lemma for fixufil 21726 and uffixfr 21727. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
uffix	|- ( ( X e. V /\ A e. X ) -> ( { { A } } e. ( fBas ` X ) /\ { x e. ~P X | A e. x } = ( X filGen { { A } } ) ) )

Distinct variable groups:   x, A   x, X   x, V

Proof of Theorem uffix

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4339	. . . 4 |- ( A e. X -> { A } C_ X )
2	1	adantl 482	. . 3 |- ( ( X e. V /\ A e. X ) -> { A } C_ X )
3		snnzg 4308	. . . 4 |- ( A e. X -> { A } =/= (/) )
4	3	adantl 482	. . 3 |- ( ( X e. V /\ A e. X ) -> { A } =/= (/) )
5		simpl 473	. . 3 |- ( ( X e. V /\ A e. X ) -> X e. V )
6		snfbas 21670	. . 3 |- ( ( { A } C_ X /\ { A } =/= (/) /\ X e. V ) -> { { A } } e. ( fBas ` X ) )
7	2, 4, 5, 6	syl3anc 1326	. 2 |- ( ( X e. V /\ A e. X ) -> { { A } } e. ( fBas ` X ) )
8		selpw 4165	. . . . . 6 |- ( y e. ~P X <-> y C_ X )
9	8	a1i 11	. . . . 5 |- ( ( X e. V /\ A e. X ) -> ( y e. ~P X <-> y C_ X ) )
10		snex 4908	. . . . . . . 8 |- { A } e. _V
11	10	snid 4208	. . . . . . 7 |- { A } e. { { A } }
12		snssi 4339	. . . . . . 7 |- ( A e. y -> { A } C_ y )
13		sseq1 3626	. . . . . . . 8 |- ( x = { A } -> ( x C_ y <-> { A } C_ y ) )
14	13	rspcev 3309	. . . . . . 7 |- ( ( { A } e. { { A } } /\ { A } C_ y ) -> E. x e. { { A } } x C_ y )
15	11, 12, 14	sylancr 695	. . . . . 6 |- ( A e. y -> E. x e. { { A } } x C_ y )
16		intss1 4492	. . . . . . . . 9 |- ( x e. { { A } } -> |^| { { A } } C_ x )
17		sstr2 3610	. . . . . . . . 9 |- ( |^| { { A } } C_ x -> ( x C_ y -> |^| { { A } } C_ y ) )
18	16, 17	syl 17	. . . . . . . 8 |- ( x e. { { A } } -> ( x C_ y -> |^| { { A } } C_ y ) )
19		snidg 4206	. . . . . . . . . . 11 |- ( A e. X -> A e. { A } )
20	19	adantl 482	. . . . . . . . . 10 |- ( ( X e. V /\ A e. X ) -> A e. { A } )
21	10	intsn 4513	. . . . . . . . . 10 |- |^| { { A } } = { A }
22	20, 21	syl6eleqr 2712	. . . . . . . . 9 |- ( ( X e. V /\ A e. X ) -> A e. |^| { { A } } )
23		ssel 3597	. . . . . . . . 9 |- ( |^| { { A } } C_ y -> ( A e. |^| { { A } } -> A e. y ) )
24	22, 23	syl5com 31	. . . . . . . 8 |- ( ( X e. V /\ A e. X ) -> ( |^| { { A } } C_ y -> A e. y ) )
25	18, 24	sylan9r 690	. . . . . . 7 |- ( ( ( X e. V /\ A e. X ) /\ x e. { { A } } ) -> ( x C_ y -> A e. y ) )
26	25	rexlimdva 3031	. . . . . 6 |- ( ( X e. V /\ A e. X ) -> ( E. x e. { { A } } x C_ y -> A e. y ) )
27	15, 26	impbid2 216	. . . . 5 |- ( ( X e. V /\ A e. X ) -> ( A e. y <-> E. x e. { { A } } x C_ y ) )
28	9, 27	anbi12d 747	. . . 4 |- ( ( X e. V /\ A e. X ) -> ( ( y e. ~P X /\ A e. y ) <-> ( y C_ X /\ E. x e. { { A } } x C_ y ) ) )
29		eleq2 2690	. . . . . 6 |- ( x = y -> ( A e. x <-> A e. y ) )
30	29	elrab 3363	. . . . 5 |- ( y e. { x e. ~P X | A e. x } <-> ( y e. ~P X /\ A e. y ) )
31	30	a1i 11	. . . 4 |- ( ( X e. V /\ A e. X ) -> ( y e. { x e. ~P X | A e. x } <-> ( y e. ~P X /\ A e. y ) ) )
32		elfg 21675	. . . . 5 |- ( { { A } } e. ( fBas ` X ) -> ( y e. ( X filGen { { A } } ) <-> ( y C_ X /\ E. x e. { { A } } x C_ y ) ) )
33	7, 32	syl 17	. . . 4 |- ( ( X e. V /\ A e. X ) -> ( y e. ( X filGen { { A } } ) <-> ( y C_ X /\ E. x e. { { A } } x C_ y ) ) )
34	28, 31, 33	3bitr4d 300	. . 3 |- ( ( X e. V /\ A e. X ) -> ( y e. { x e. ~P X | A e. x } <-> y e. ( X filGen { { A } } ) ) )
35	34	eqrdv 2620	. 2 |- ( ( X e. V /\ A e. X ) -> { x e. ~P X | A e. x } = ( X filGen { { A } } ) )
36	7, 35	jca 554	1 |- ( ( X e. V /\ A e. X ) -> ( { { A } } e. ( fBas ` X ) /\ { x e. ~P X | A e. x } = ( X filGen { { A } } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735

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

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-nel 2898  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-int 4476  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-fbas 19743  df-fg 19744  df-fil 21650

This theorem is referenced by:  fixufil  21726  uffixfr  21727