Theorem uffix 20588

Description: Lemma for fixufil 20589 and uffixfr 20590. (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 4160	. . . 4 |- ( A e. X -> { A } C_ X )
2	1	adantl 464	. . 3 |- ( ( X e. V /\ A e. X ) -> { A } C_ X )
3		snnzg 4133	. . . 4 |- ( A e. X -> { A } =/= (/) )
4	3	adantl 464	. . 3 |- ( ( X e. V /\ A e. X ) -> { A } =/= (/) )
5		simpl 455	. . 3 |- ( ( X e. V /\ A e. X ) -> X e. V )
6		snfbas 20533	. . 3 |- ( ( { A } C_ X /\ { A } =/= (/) /\ X e. V ) -> { { A } } e. ( fBas ` X ) )
7	2, 4, 5, 6	syl3anc 1226	. 2 |- ( ( X e. V /\ A e. X ) -> { { A } } e. ( fBas ` X ) )
8		selpw 4006	. . . . . 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 4678	. . . . . . . 8 |- { A } e. _V
11	10	snid 4044	. . . . . . 7 |- { A } e. { { A } }
12		snssi 4160	. . . . . . 7 |- ( A e. y -> { A } C_ y )
13		sseq1 3510	. . . . . . . 8 |- ( x = { A } -> ( x C_ y <-> { A } C_ y ) )
14	13	rspcev 3207	. . . . . . 7 |- ( ( { A } e. { { A } } /\ { A } C_ y ) -> E. x e. { { A } } x C_ y )
15	11, 12, 14	sylancr 661	. . . . . 6 |- ( A e. y -> E. x e. { { A } } x C_ y )
16		intss1 4286	. . . . . . . . 9 |- ( x e. { { A } } -> |^| { { A } } C_ x )
17		sstr2 3496	. . . . . . . . 9 |- ( |^| { { A } } C_ x -> ( x C_ y -> |^| { { A } } C_ y ) )
18	16, 17	syl 16	. . . . . . . 8 |- ( x e. { { A } } -> ( x C_ y -> |^| { { A } } C_ y ) )
19		snidg 4042	. . . . . . . . . . 11 |- ( A e. X -> A e. { A } )
20	19	adantl 464	. . . . . . . . . 10 |- ( ( X e. V /\ A e. X ) -> A e. { A } )
21	10	intsn 4308	. . . . . . . . . 10 |- |^| { { A } } = { A }
22	20, 21	syl6eleqr 2553	. . . . . . . . 9 |- ( ( X e. V /\ A e. X ) -> A e. |^| { { A } } )
23		ssel 3483	. . . . . . . . 9 |- ( |^| { { A } } C_ y -> ( A e. |^| { { A } } -> A e. y ) )
24	22, 23	syl5com 30	. . . . . . . 8 |- ( ( X e. V /\ A e. X ) -> ( |^| { { A } } C_ y -> A e. y ) )
25	18, 24	sylan9r 656	. . . . . . 7 |- ( ( ( X e. V /\ A e. X ) /\ x e. { { A } } ) -> ( x C_ y -> A e. y ) )
26	25	rexlimdva 2946	. . . . . 6 |- ( ( X e. V /\ A e. X ) -> ( E. x e. { { A } } x C_ y -> A e. y ) )
27	15, 26	impbid2 204	. . . . 5 |- ( ( X e. V /\ A e. X ) -> ( A e. y <-> E. x e. { { A } } x C_ y ) )
28	9, 27	anbi12d 708	. . . 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 2527	. . . . . 6 |- ( x = y -> ( A e. x <-> A e. y ) )
30	29	elrab 3254	. . . . 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 20538	. . . . 5 |- ( { { A } } e. ( fBas ` X ) -> ( y e. ( X filGen { { A } } ) <-> ( y C_ X /\ E. x e. { { A } } x C_ y ) ) )
33	7, 32	syl 16	. . . 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 285	. . 3 |- ( ( X e. V /\ A e. X ) -> ( y e. { x e. ~P X | A e. x } <-> y e. ( X filGen { { A } } ) ) )
35	34	eqrdv 2451	. 2 |- ( ( X e. V /\ A e. X ) -> { x e. ~P X | A e. x } = ( X filGen { { A } } ) )
36	7, 35	jca 530	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 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   {crab 2808   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   |^|cint 4271   ` cfv 5570  (class class class)co 6270   fBascfbas 18601   filGencfg 18602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-fbas 18611  df-fg 18612  df-fil 20513

This theorem is referenced by:  fixufil  20589  uffixfr  20590