Theorem rexsupp 5758

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 5753	. . . . 5 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
2		eldifsn 3794	. . . . . . 7 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) )
3		funfvex 5643	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5422	. . . . . . . 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 2533	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 2200   =/= wne 2400   E.wrex 2509   _Vcvv 2799   \ cdif 3194   {csn 3666   `'ccnv 4717   "cima 4721   Fn wfn 5312   ` cfv 5317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by: (None)