Theorem elrfirn 37258

Description: Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
elrfirn	|- ( ( B e. V /\ F : I --> ~P B ) -> ( A e. ( fi ` ( { B } u. ran F ) ) <-> E. v e. ( ~P I i^i Fin ) A = ( B i^i |^|_ y e. v ( F ` y ) ) ) )

Distinct variable groups:   v, A   v, B   v, F, y   v, I   v, V   y, v

Allowed substitution hints:   A( y)   B( y)   I( y)   V( y)

Proof of Theorem elrfirn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frn 6053	. . 3 |- ( F : I --> ~P B -> ran F C_ ~P B )
2		elrfi 37257	. . 3 |- ( ( B e. V /\ ran F C_ ~P B ) -> ( A e. ( fi ` ( { B } u. ran F ) ) <-> E. w e. ( ~P ran F i^i Fin ) A = ( B i^i |^| w ) ) )
3	1, 2	sylan2 491	. 2 |- ( ( B e. V /\ F : I --> ~P B ) -> ( A e. ( fi ` ( { B } u. ran F ) ) <-> E. w e. ( ~P ran F i^i Fin ) A = ( B i^i |^| w ) ) )
4		imassrn 5477	. . . . . 6 |- ( F " v ) C_ ran F
5		pwexg 4850	. . . . . . . 8 |- ( B e. V -> ~P B e. _V )
6		ssexg 4804	. . . . . . . 8 |- ( ( ran F C_ ~P B /\ ~P B e. _V ) -> ran F e. _V )
7	1, 5, 6	syl2anr 495	. . . . . . 7 |- ( ( B e. V /\ F : I --> ~P B ) -> ran F e. _V )
8		elpw2g 4827	. . . . . . 7 |- ( ran F e. _V -> ( ( F " v ) e. ~P ran F <-> ( F " v ) C_ ran F ) )
9	7, 8	syl 17	. . . . . 6 |- ( ( B e. V /\ F : I --> ~P B ) -> ( ( F " v ) e. ~P ran F <-> ( F " v ) C_ ran F ) )
10	4, 9	mpbiri 248	. . . . 5 |- ( ( B e. V /\ F : I --> ~P B ) -> ( F " v ) e. ~P ran F )
11	10	adantr 481	. . . 4 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> ( F " v ) e. ~P ran F )
12		ffun 6048	. . . . . 6 |- ( F : I --> ~P B -> Fun F )
13	12	ad2antlr 763	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> Fun F )
14		inss2 3834	. . . . . . 7 |- ( ~P I i^i Fin ) C_ Fin
15	14	sseli 3599	. . . . . 6 |- ( v e. ( ~P I i^i Fin ) -> v e. Fin )
16	15	adantl 482	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> v e. Fin )
17		imafi 8259	. . . . 5 |- ( ( Fun F /\ v e. Fin ) -> ( F " v ) e. Fin )
18	13, 16, 17	syl2anc 693	. . . 4 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> ( F " v ) e. Fin )
19	11, 18	elind 3798	. . 3 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> ( F " v ) e. ( ~P ran F i^i Fin ) )
20		ffn 6045	. . . . . 6 |- ( F : I --> ~P B -> F Fn I )
21	20	ad2antlr 763	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w e. ( ~P ran F i^i Fin ) ) -> F Fn I )
22		inss1 3833	. . . . . . . 8 |- ( ~P ran F i^i Fin ) C_ ~P ran F
23	22	sseli 3599	. . . . . . 7 |- ( w e. ( ~P ran F i^i Fin ) -> w e. ~P ran F )
24	23	elpwid 4170	. . . . . 6 |- ( w e. ( ~P ran F i^i Fin ) -> w C_ ran F )
25	24	adantl 482	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w e. ( ~P ran F i^i Fin ) ) -> w C_ ran F )
26		inss2 3834	. . . . . . 7 |- ( ~P ran F i^i Fin ) C_ Fin
27	26	sseli 3599	. . . . . 6 |- ( w e. ( ~P ran F i^i Fin ) -> w e. Fin )
28	27	adantl 482	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w e. ( ~P ran F i^i Fin ) ) -> w e. Fin )
29		fipreima 8272	. . . . 5 |- ( ( F Fn I /\ w C_ ran F /\ w e. Fin ) -> E. v e. ( ~P I i^i Fin ) ( F " v ) = w )
30	21, 25, 28, 29	syl3anc 1326	. . . 4 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w e. ( ~P ran F i^i Fin ) ) -> E. v e. ( ~P I i^i Fin ) ( F " v ) = w )
31		eqcom 2629	. . . . 5 |- ( ( F " v ) = w <-> w = ( F " v ) )
32	31	rexbii 3041	. . . 4 |- ( E. v e. ( ~P I i^i Fin ) ( F " v ) = w <-> E. v e. ( ~P I i^i Fin ) w = ( F " v ) )
33	30, 32	sylib 208	. . 3 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w e. ( ~P ran F i^i Fin ) ) -> E. v e. ( ~P I i^i Fin ) w = ( F " v ) )
34		inteq 4478	. . . . . 6 |- ( w = ( F " v ) -> |^| w = |^| ( F " v ) )
35	34	ineq2d 3814	. . . . 5 |- ( w = ( F " v ) -> ( B i^i |^| w ) = ( B i^i |^| ( F " v ) ) )
36	35	eqeq2d 2632	. . . 4 |- ( w = ( F " v ) -> ( A = ( B i^i |^| w ) <-> A = ( B i^i |^| ( F " v ) ) ) )
37	36	adantl 482	. . 3 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ w = ( F " v ) ) -> ( A = ( B i^i |^| w ) <-> A = ( B i^i |^| ( F " v ) ) ) )
38	19, 33, 37	rexxfrd 4881	. 2 |- ( ( B e. V /\ F : I --> ~P B ) -> ( E. w e. ( ~P ran F i^i Fin ) A = ( B i^i |^| w ) <-> E. v e. ( ~P I i^i Fin ) A = ( B i^i |^| ( F " v ) ) ) )
39	20	ad2antlr 763	. . . . . . 7 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> F Fn I )
40		inss1 3833	. . . . . . . . . 10 |- ( ~P I i^i Fin ) C_ ~P I
41	40	sseli 3599	. . . . . . . . 9 |- ( v e. ( ~P I i^i Fin ) -> v e. ~P I )
42	41	elpwid 4170	. . . . . . . 8 |- ( v e. ( ~P I i^i Fin ) -> v C_ I )
43	42	adantl 482	. . . . . . 7 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> v C_ I )
44		imaiinfv 37256	. . . . . . 7 |- ( ( F Fn I /\ v C_ I ) -> |^|_ y e. v ( F ` y ) = |^| ( F " v ) )
45	39, 43, 44	syl2anc 693	. . . . . 6 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> |^|_ y e. v ( F ` y ) = |^| ( F " v ) )
46	45	eqcomd 2628	. . . . 5 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> |^| ( F " v ) = |^|_ y e. v ( F ` y ) )
47	46	ineq2d 3814	. . . 4 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> ( B i^i |^| ( F " v ) ) = ( B i^i |^|_ y e. v ( F ` y ) ) )
48	47	eqeq2d 2632	. . 3 |- ( ( ( B e. V /\ F : I --> ~P B ) /\ v e. ( ~P I i^i Fin ) ) -> ( A = ( B i^i |^| ( F " v ) ) <-> A = ( B i^i |^|_ y e. v ( F ` y ) ) ) )
49	48	rexbidva 3049	. 2 |- ( ( B e. V /\ F : I --> ~P B ) -> ( E. v e. ( ~P I i^i Fin ) A = ( B i^i |^| ( F " v ) ) <-> E. v e. ( ~P I i^i Fin ) A = ( B i^i |^|_ y e. v ( F ` y ) ) ) )
50	3, 38, 49	3bitrd 294	1 |- ( ( B e. V /\ F : I --> ~P B ) -> ( A e. ( fi ` ( { B } u. ran F ) ) <-> E. v e. ( ~P I i^i Fin ) A = ( B i^i |^|_ y e. v ( F ` y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   |^|cint 4475   |^|_ciin 4521   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` 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-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317

This theorem is referenced by:  elrfirn2  37259