Theorem rexsupp 5423

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 5418	. . . . 5 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
2		funfvex 5322	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
3	2	funfni 5114	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
4	3	biantrurd 299	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) =/= Z <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) ) )
5		eldifsn 3567	. . . . . . 7 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) )
6	4, 5	syl6rbbr 197	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. ( _V \ { Z } ) <-> ( F ` x ) =/= Z ) )
7	6	pm5.32da 440	. . . . 5 |- ( F Fn A -> ( ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
8	1, 7	bitrd 186	. . . 4 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
9	8	anbi1d 453	. . 3 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) ) )
10		anass 393	. . 3 |- ( ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) )
11	9, 10	syl6bb 194	. 2 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) ) )
12	11	rexbidv2 2383	1 |- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <-> E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1438   =/= wne 2255   E.wrex 2360   _Vcvv 2619   \ cdif 2996   {csn 3446   `'ccnv 4437   "cima 4441   Fn wfn 5010   ` cfv 5015

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023

This theorem is referenced by: (None)