Theorem isf34lem2 9195

Description: Lemma for isfin3-4 9204. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )

Assertion

Ref	Expression
isf34lem2	|- ( A e. V -> F : ~P A --> ~P A )

Distinct variable groups:   x, A   x, V

Allowed substitution hint:   F( x)

Proof of Theorem isf34lem2

Step	Hyp	Ref	Expression
1		difss 3737	. . . 4 |- ( A \ x ) C_ A
2		elpw2g 4827	. . . 4 |- ( A e. V -> ( ( A \ x ) e. ~P A <-> ( A \ x ) C_ A ) )
3	1, 2	mpbiri 248	. . 3 |- ( A e. V -> ( A \ x ) e. ~P A )
4	3	adantr 481	. 2 |- ( ( A e. V /\ x e. ~P A ) -> ( A \ x ) e. ~P A )
5		compss.a	. 2 |- F = ( x e. ~P A |-> ( A \ x ) )
6	4, 5	fmptd 6385	1 |- ( A e. V -> F : ~P A --> ~P A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   -->wf 5884

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  isf34lem5  9200  isf34lem7  9201  isf34lem6  9202