Theorem elfi2 8320

Description: The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
elfi2	|- ( B e. V -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )

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

Proof of Theorem elfi2

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( A e. ( fi ` B ) -> A e. _V )
2	1	a1i 11	. 2 |- ( B e. V -> ( A e. ( fi ` B ) -> A e. _V ) )
3		simpr 477	. . . . 5 |- ( ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) -> A = |^| x )
4		eldifsni 4320	. . . . . . 7 |- ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) -> x =/= (/) )
5	4	adantr 481	. . . . . 6 |- ( ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) -> x =/= (/) )
6		intex 4820	. . . . . 6 |- ( x =/= (/) <-> |^| x e. _V )
7	5, 6	sylib 208	. . . . 5 |- ( ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) -> |^| x e. _V )
8	3, 7	eqeltrd 2701	. . . 4 |- ( ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) -> A e. _V )
9	8	rexlimiva 3028	. . 3 |- ( E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x -> A e. _V )
10	9	a1i 11	. 2 |- ( B e. V -> ( E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x -> A e. _V ) )
11		elfi 8319	. . . 4 |- ( ( A e. _V /\ B e. V ) -> ( A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )
12		vprc 4796	. . . . . . . . . . 11 |- -. _V e. _V
13		elsni 4194	. . . . . . . . . . . . . 14 |- ( x e. { (/) } -> x = (/) )
14	13	inteqd 4480	. . . . . . . . . . . . 13 |- ( x e. { (/) } -> |^| x = |^| (/) )
15		int0 4490	. . . . . . . . . . . . 13 |- |^| (/) = _V
16	14, 15	syl6eq 2672	. . . . . . . . . . . 12 |- ( x e. { (/) } -> |^| x = _V )
17	16	eleq1d 2686	. . . . . . . . . . 11 |- ( x e. { (/) } -> ( |^| x e. _V <-> _V e. _V ) )
18	12, 17	mtbiri 317	. . . . . . . . . 10 |- ( x e. { (/) } -> -. |^| x e. _V )
19		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> A = |^| x )
20		simpll 790	. . . . . . . . . . 11 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> A e. _V )
21	19, 20	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> |^| x e. _V )
22	18, 21	nsyl3 133	. . . . . . . . 9 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> -. x e. { (/) } )
23	22	biantrud 528	. . . . . . . 8 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> ( x e. ( ~P B i^i Fin ) <-> ( x e. ( ~P B i^i Fin ) /\ -. x e. { (/) } ) ) )
24		eldif 3584	. . . . . . . 8 |- ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) <-> ( x e. ( ~P B i^i Fin ) /\ -. x e. { (/) } ) )
25	23, 24	syl6bbr 278	. . . . . . 7 |- ( ( ( A e. _V /\ B e. V ) /\ A = |^| x ) -> ( x e. ( ~P B i^i Fin ) <-> x e. ( ( ~P B i^i Fin ) \ { (/) } ) ) )
26	25	pm5.32da 673	. . . . . 6 |- ( ( A e. _V /\ B e. V ) -> ( ( A = |^| x /\ x e. ( ~P B i^i Fin ) ) <-> ( A = |^| x /\ x e. ( ( ~P B i^i Fin ) \ { (/) } ) ) ) )
27		ancom 466	. . . . . 6 |- ( ( x e. ( ~P B i^i Fin ) /\ A = |^| x ) <-> ( A = |^| x /\ x e. ( ~P B i^i Fin ) ) )
28		ancom 466	. . . . . 6 |- ( ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) <-> ( A = |^| x /\ x e. ( ( ~P B i^i Fin ) \ { (/) } ) ) )
29	26, 27, 28	3bitr4g 303	. . . . 5 |- ( ( A e. _V /\ B e. V ) -> ( ( x e. ( ~P B i^i Fin ) /\ A = |^| x ) <-> ( x e. ( ( ~P B i^i Fin ) \ { (/) } ) /\ A = |^| x ) ) )
30	29	rexbidv2 3048	. . . 4 |- ( ( A e. _V /\ B e. V ) -> ( E. x e. ( ~P B i^i Fin ) A = |^| x <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )
31	11, 30	bitrd 268	. . 3 |- ( ( A e. _V /\ B e. V ) -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )
32	31	expcom 451	. 2 |- ( B e. V -> ( A e. _V -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) ) )
33	2, 10, 32	pm5.21ndd 369	1 |- ( B e. V -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|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:  fifo  8338  firest  16093  alexsublem  21848  ispisys2  30216