Theorem inffien 8886

Description: The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
inffien	|- ( ( A e. dom card /\ om ~<_ A ) -> ( fi ` A ) ~~ A )

Proof of Theorem inffien

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infpwfien 8885	. . . . . . . 8 |- ( ( A e. dom card /\ om ~<_ A ) -> ( ~P A i^i Fin ) ~~ A )
2		relen 7960	. . . . . . . . 9 |- Rel ~~
3	2	brrelexi 5158	. . . . . . . 8 |- ( ( ~P A i^i Fin ) ~~ A -> ( ~P A i^i Fin ) e. _V )
4	1, 3	syl 17	. . . . . . 7 |- ( ( A e. dom card /\ om ~<_ A ) -> ( ~P A i^i Fin ) e. _V )
5		difss 3737	. . . . . . 7 |- ( ( ~P A i^i Fin ) \ { (/) } ) C_ ( ~P A i^i Fin )
6		ssdomg 8001	. . . . . . 7 |- ( ( ~P A i^i Fin ) e. _V -> ( ( ( ~P A i^i Fin ) \ { (/) } ) C_ ( ~P A i^i Fin ) -> ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ ( ~P A i^i Fin ) ) )
7	4, 5, 6	mpisyl 21	. . . . . 6 |- ( ( A e. dom card /\ om ~<_ A ) -> ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ ( ~P A i^i Fin ) )
8		domentr 8015	. . . . . 6 |- ( ( ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ ( ~P A i^i Fin ) /\ ( ~P A i^i Fin ) ~~ A ) -> ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ A )
9	7, 1, 8	syl2anc 693	. . . . 5 |- ( ( A e. dom card /\ om ~<_ A ) -> ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ A )
10		numdom 8861	. . . . 5 |- ( ( A e. dom card /\ ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ A ) -> ( ( ~P A i^i Fin ) \ { (/) } ) e. dom card )
11	9, 10	syldan 487	. . . 4 |- ( ( A e. dom card /\ om ~<_ A ) -> ( ( ~P A i^i Fin ) \ { (/) } ) e. dom card )
12		eqid 2622	. . . . . 6 |- ( x e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| x ) = ( x e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| x )
13	12	fifo 8338	. . . . 5 |- ( A e. dom card -> ( x e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| x ) : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )
14	13	adantr 481	. . . 4 |- ( ( A e. dom card /\ om ~<_ A ) -> ( x e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| x ) : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )
15		fodomnum 8880	. . . 4 |- ( ( ( ~P A i^i Fin ) \ { (/) } ) e. dom card -> ( ( x e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| x ) : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) -> ( fi ` A ) ~<_ ( ( ~P A i^i Fin ) \ { (/) } ) ) )
16	11, 14, 15	sylc 65	. . 3 |- ( ( A e. dom card /\ om ~<_ A ) -> ( fi ` A ) ~<_ ( ( ~P A i^i Fin ) \ { (/) } ) )
17		domtr 8009	. . 3 |- ( ( ( fi ` A ) ~<_ ( ( ~P A i^i Fin ) \ { (/) } ) /\ ( ( ~P A i^i Fin ) \ { (/) } ) ~<_ A ) -> ( fi ` A ) ~<_ A )
18	16, 9, 17	syl2anc 693	. 2 |- ( ( A e. dom card /\ om ~<_ A ) -> ( fi ` A ) ~<_ A )
19		fvex 6201	. . 3 |- ( fi ` A ) e. _V
20		ssfii 8325	. . . 4 |- ( A e. dom card -> A C_ ( fi ` A ) )
21	20	adantr 481	. . 3 |- ( ( A e. dom card /\ om ~<_ A ) -> A C_ ( fi ` A ) )
22		ssdomg 8001	. . 3 |- ( ( fi ` A ) e. _V -> ( A C_ ( fi ` A ) -> A ~<_ ( fi ` A ) ) )
23	19, 21, 22	mpsyl 68	. 2 |- ( ( A e. dom card /\ om ~<_ A ) -> A ~<_ ( fi ` A ) )
24		sbth 8080	. 2 |- ( ( ( fi ` A ) ~<_ A /\ A ~<_ ( fi ` A ) ) -> ( fi ` A ) ~~ A )
25	18, 23, 24	syl2anc 693	1 |- ( ( A e. dom card /\ om ~<_ A ) -> ( fi ` A ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -onto->wfo 5886   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   ficfi 8316   cardccrd 8761

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-rmo 2920  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-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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-oi 8415  df-card 8765  df-acn 8768

This theorem is referenced by: (None)