Theorem rexsupp 5682

Description: Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Assertion

Ref	Expression
rexsupp	|- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <-> E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) )

Distinct variable groups:   x, F   x, A

Allowed substitution hints:   ph( x)   Z( x)

Proof of Theorem rexsupp

Step	Hyp	Ref	Expression
1		elpreima 5677	. . . . 5 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
2		eldifsn 3745	. . . . . . 7 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) )
3		funfvex 5571	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5354	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
5	4	biantrurd 305	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) =/= Z <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) ) )
6	2, 5	bitr4id 199	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. ( _V \ { Z } ) <-> ( F ` x ) =/= Z ) )
7	6	pm5.32da 452	. . . . 5 |- ( F Fn A -> ( ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
8	1, 7	bitrd 188	. . . 4 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
9	8	anbi1d 465	. . 3 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) ) )
10		anass 401	. . 3 |- ( ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) )
11	9, 10	bitrdi 196	. 2 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) ) )
12	11	rexbidv2 2497	1 |- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <-> E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   =/= wne 2364   E.wrex 2473   _Vcvv 2760   \ cdif 3150   {csn 3618   `'ccnv 4658   "cima 4662   Fn wfn 5249   ` cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by: (None)