Theorem elfi 8319

Description: Specific properties of an element of ( fi ` B ). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

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

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

Proof of Theorem elfi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fival 8318	. . 3 |- ( B e. W -> ( fi ` B ) = { y | E. x e. ( ~P B i^i Fin ) y = |^| x } )
2	1	eleq2d 2687	. 2 |- ( B e. W -> ( A e. ( fi ` B ) <-> A e. { y | E. x e. ( ~P B i^i Fin ) y = |^| x } ) )
3		eqeq1 2626	. . . 4 |- ( y = A -> ( y = |^| x <-> A = |^| x ) )
4	3	rexbidv 3052	. . 3 |- ( y = A -> ( E. x e. ( ~P B i^i Fin ) y = |^| x <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )
5	4	elabg 3351	. 2 |- ( A e. V -> ( A e. { y | E. x e. ( ~P B i^i Fin ) y = |^| x } <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )
6	2, 5	sylan9bbr 737	1 |- ( ( A e. V /\ B e. W ) -> ( A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   i^i cin 3573   ~Pcpw 4158   |^|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:  elfi2  8320  elfir  8321  inelfi  8324  fiin  8328  dffi2  8329  elfiun  8336  subbascn  21058  cmpfi  21211  fbasfip  21672  alexsubALTlem4  21854  heibor1lem  33608  elrfi  37257