Theorem fifo 8338

Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)

Hypothesis

Ref	Expression
fifo.1	|- F = ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| y )

Assertion

Ref	Expression
fifo	|- ( A e. V -> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )

Distinct variable groups:   y, A   y, V

Allowed substitution hint:   F( y)

Proof of Theorem fifo

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsni 4320	. . . . . 6 |- ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) -> y =/= (/) )
2		intex 4820	. . . . . 6 |- ( y =/= (/) <-> |^| y e. _V )
3	1, 2	sylib 208	. . . . 5 |- ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) -> |^| y e. _V )
4	3	rgen 2922	. . . 4 |- A. y e. ( ( ~P A i^i Fin ) \ { (/) } ) |^| y e. _V
5		fifo.1	. . . . 5 |- F = ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| y )
6	5	fnmpt 6020	. . . 4 |- ( A. y e. ( ( ~P A i^i Fin ) \ { (/) } ) |^| y e. _V -> F Fn ( ( ~P A i^i Fin ) \ { (/) } ) )
7	4, 6	mp1i 13	. . 3 |- ( A e. V -> F Fn ( ( ~P A i^i Fin ) \ { (/) } ) )
8		dffn4 6121	. . 3 |- ( F Fn ( ( ~P A i^i Fin ) \ { (/) } ) <-> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ran F )
9	7, 8	sylib 208	. 2 |- ( A e. V -> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ran F )
10		elfi2 8320	. . . . 5 |- ( A e. V -> ( x e. ( fi ` A ) <-> E. y e. ( ( ~P A i^i Fin ) \ { (/) } ) x = |^| y ) )
11		vex 3203	. . . . . 6 |- x e. _V
12	5	elrnmpt 5372	. . . . . 6 |- ( x e. _V -> ( x e. ran F <-> E. y e. ( ( ~P A i^i Fin ) \ { (/) } ) x = |^| y ) )
13	11, 12	ax-mp 5	. . . . 5 |- ( x e. ran F <-> E. y e. ( ( ~P A i^i Fin ) \ { (/) } ) x = |^| y )
14	10, 13	syl6bbr 278	. . . 4 |- ( A e. V -> ( x e. ( fi ` A ) <-> x e. ran F ) )
15	14	eqrdv 2620	. . 3 |- ( A e. V -> ( fi ` A ) = ran F )
16		foeq3 6113	. . 3 |- ( ( fi ` A ) = ran F -> ( F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) <-> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ran F ) )
17	15, 16	syl 17	. 2 |- ( A e. V -> ( F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) <-> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ran F ) )
18	9, 17	mpbird 247	1 |- ( A e. V -> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ` 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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fo 5894  df-fv 5896  df-fi 8317

This theorem is referenced by:  inffien  8886  fictb  9067