Theorem rexsupp 5620

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 5615	. . . . 5 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) ) )
2		eldifsn 3710	. . . . . . 7 |- ( ( F ` x ) e. ( _V \ { Z } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) )
3		funfvex 5513	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5298	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
5	4	biantrurd 303	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) =/= Z <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= Z ) ) )
6	2, 5	bitr4id 198	. . . . . 6 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. ( _V \ { Z } ) <-> ( F ` x ) =/= Z ) )
7	6	pm5.32da 449	. . . . 5 |- ( F Fn A -> ( ( x e. A /\ ( F ` x ) e. ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
8	1, 7	bitrd 187	. . . 4 |- ( F Fn A -> ( x e. ( `' F " ( _V \ { Z } ) ) <-> ( x e. A /\ ( F ` x ) =/= Z ) ) )
9	8	anbi1d 462	. . 3 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) ) )
10		anass 399	. . 3 |- ( ( ( x e. A /\ ( F ` x ) =/= Z ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) )
11	9, 10	bitrdi 195	. 2 |- ( F Fn A -> ( ( x e. ( `' F " ( _V \ { Z } ) ) /\ ph ) <-> ( x e. A /\ ( ( F ` x ) =/= Z /\ ph ) ) ) )
12	11	rexbidv2 2473	1 |- ( F Fn A -> ( E. x e. ( `' F " ( _V \ { Z } ) ) ph <-> E. x e. A ( ( F ` x ) =/= Z /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   =/= wne 2340   E.wrex 2449   _Vcvv 2730   \ cdif 3118   {csn 3583   `'ccnv 4610   "cima 4614   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)