Theorem fival 8318

Description: The set of all the finite intersections of the elements of A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fival	|- ( A e. V -> ( fi ` A ) = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )

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

Allowed substitution hint:   V( y)

Proof of Theorem fival

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		simpr 477	. . . . . . 7 |- ( ( x e. ( ~P A i^i Fin ) /\ y = |^| x ) -> y = |^| x )
3		inss1 3833	. . . . . . . . . 10 |- ( ~P A i^i Fin ) C_ ~P A
4	3	sseli 3599	. . . . . . . . 9 |- ( x e. ( ~P A i^i Fin ) -> x e. ~P A )
5	4	elpwid 4170	. . . . . . . 8 |- ( x e. ( ~P A i^i Fin ) -> x C_ A )
6		eqvisset 3211	. . . . . . . . 9 |- ( y = |^| x -> |^| x e. _V )
7		intex 4820	. . . . . . . . 9 |- ( x =/= (/) <-> |^| x e. _V )
8	6, 7	sylibr 224	. . . . . . . 8 |- ( y = |^| x -> x =/= (/) )
9		intssuni2 4502	. . . . . . . 8 |- ( ( x C_ A /\ x =/= (/) ) -> |^| x C_ U. A )
10	5, 8, 9	syl2an 494	. . . . . . 7 |- ( ( x e. ( ~P A i^i Fin ) /\ y = |^| x ) -> |^| x C_ U. A )
11	2, 10	eqsstrd 3639	. . . . . 6 |- ( ( x e. ( ~P A i^i Fin ) /\ y = |^| x ) -> y C_ U. A )
12		selpw 4165	. . . . . 6 |- ( y e. ~P U. A <-> y C_ U. A )
13	11, 12	sylibr 224	. . . . 5 |- ( ( x e. ( ~P A i^i Fin ) /\ y = |^| x ) -> y e. ~P U. A )
14	13	rexlimiva 3028	. . . 4 |- ( E. x e. ( ~P A i^i Fin ) y = |^| x -> y e. ~P U. A )
15	14	abssi 3677	. . 3 |- { y | E. x e. ( ~P A i^i Fin ) y = |^| x } C_ ~P U. A
16		uniexg 6955	. . . 4 |- ( A e. V -> U. A e. _V )
17		pwexg 4850	. . . 4 |- ( U. A e. _V -> ~P U. A e. _V )
18	16, 17	syl 17	. . 3 |- ( A e. V -> ~P U. A e. _V )
19		ssexg 4804	. . 3 |- ( ( { y | E. x e. ( ~P A i^i Fin ) y = |^| x } C_ ~P U. A /\ ~P U. A e. _V ) -> { y | E. x e. ( ~P A i^i Fin ) y = |^| x } e. _V )
20	15, 18, 19	sylancr 695	. 2 |- ( A e. V -> { y | E. x e. ( ~P A i^i Fin ) y = |^| x } e. _V )
21		pweq 4161	. . . . . 6 |- ( z = A -> ~P z = ~P A )
22	21	ineq1d 3813	. . . . 5 |- ( z = A -> ( ~P z i^i Fin ) = ( ~P A i^i Fin ) )
23	22	rexeqdv 3145	. . . 4 |- ( z = A -> ( E. x e. ( ~P z i^i Fin ) y = |^| x <-> E. x e. ( ~P A i^i Fin ) y = |^| x ) )
24	23	abbidv 2741	. . 3 |- ( z = A -> { y | E. x e. ( ~P z i^i Fin ) y = |^| x } = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )
25		df-fi 8317	. . 3 |- fi = ( z e. _V |-> { y | E. x e. ( ~P z i^i Fin ) y = |^| x } )
26	24, 25	fvmptg 6280	. 2 |- ( ( A e. _V /\ { y | E. x e. ( ~P A i^i Fin ) y = |^| x } e. _V ) -> ( fi ` A ) = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )
27	1, 20, 26	syl2anc 693	1 |- ( A e. V -> ( fi ` A ) = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ` cfv 5888   Fincfn 7955   ficfi 8316

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-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-iota 5851  df-fun 5890  df-fv 5896  df-fi 8317

This theorem is referenced by:  elfi  8319  fi0  8326