Theorem inelfi 8324

Description: The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)

Assertion

Ref	Expression
inelfi	|- ( ( X e. V /\ A e. X /\ B e. X ) -> ( A i^i B ) e. ( fi ` X ) )

Proof of Theorem inelfi

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prelpwi 4915	. . . . 5 |- ( ( A e. X /\ B e. X ) -> { A , B } e. ~P X )
2	1	3adant1 1079	. . . 4 |- ( ( X e. V /\ A e. X /\ B e. X ) -> { A , B } e. ~P X )
3		prfi 8235	. . . . 5 |- { A , B } e. Fin
4	3	a1i 11	. . . 4 |- ( ( X e. V /\ A e. X /\ B e. X ) -> { A , B } e. Fin )
5	2, 4	elind 3798	. . 3 |- ( ( X e. V /\ A e. X /\ B e. X ) -> { A , B } e. ( ~P X i^i Fin ) )
6		intprg 4511	. . . . 5 |- ( ( A e. X /\ B e. X ) -> |^| { A , B } = ( A i^i B ) )
7	6	3adant1 1079	. . . 4 |- ( ( X e. V /\ A e. X /\ B e. X ) -> |^| { A , B } = ( A i^i B ) )
8	7	eqcomd 2628	. . 3 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( A i^i B ) = |^| { A , B } )
9		inteq 4478	. . . . 5 |- ( p = { A , B } -> |^| p = |^| { A , B } )
10	9	eqeq2d 2632	. . . 4 |- ( p = { A , B } -> ( ( A i^i B ) = |^| p <-> ( A i^i B ) = |^| { A , B } ) )
11	10	rspcev 3309	. . 3 |- ( ( { A , B } e. ( ~P X i^i Fin ) /\ ( A i^i B ) = |^| { A , B } ) -> E. p e. ( ~P X i^i Fin ) ( A i^i B ) = |^| p )
12	5, 8, 11	syl2anc 693	. 2 |- ( ( X e. V /\ A e. X /\ B e. X ) -> E. p e. ( ~P X i^i Fin ) ( A i^i B ) = |^| p )
13		inex1g 4801	. . . 4 |- ( A e. X -> ( A i^i B ) e. _V )
14	13	3ad2ant2 1083	. . 3 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( A i^i B ) e. _V )
15		simp1 1061	. . 3 |- ( ( X e. V /\ A e. X /\ B e. X ) -> X e. V )
16		elfi 8319	. . 3 |- ( ( ( A i^i B ) e. _V /\ X e. V ) -> ( ( A i^i B ) e. ( fi ` X ) <-> E. p e. ( ~P X i^i Fin ) ( A i^i B ) = |^| p ) )
17	14, 15, 16	syl2anc 693	. 2 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( ( A i^i B ) e. ( fi ` X ) <-> E. p e. ( ~P X i^i Fin ) ( A i^i B ) = |^| p ) )
18	12, 17	mpbird 247	1 |- ( ( X e. V /\ A e. X /\ B e. X ) -> ( A i^i B ) e. ( fi ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   {cpr 4179   |^|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-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-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-fin 7959  df-fi 8317

This theorem is referenced by:  neiptoptop  20935  sigapildsyslem  30224