Theorem rexsupp 7313

Description: Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)

Assertion

Ref	Expression
rexsupp	|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( E. x e. ( F supp Z ) ph <-> E. x e. X ( ( F ` x ) =/= Z /\ ph ) ) )

Distinct variable groups:   x, F   x, V   x, W   x, X   x, Z

Allowed substitution hint:   ph( x)

Proof of Theorem rexsupp

Step	Hyp	Ref	Expression
1		elsuppfn 7303	. . . 4 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> ( x e. X /\ ( F ` x ) =/= Z ) ) )
2	1	anbi1d 741	. . 3 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( ( x e. ( F supp Z ) /\ ph ) <-> ( ( x e. X /\ ( F ` x ) =/= Z ) /\ ph ) ) )
3		anass 681	. . 3 |- ( ( ( x e. X /\ ( F ` x ) =/= Z ) /\ ph ) <-> ( x e. X /\ ( ( F ` x ) =/= Z /\ ph ) ) )
4	2, 3	syl6bb 276	. 2 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( ( x e. ( F supp Z ) /\ ph ) <-> ( x e. X /\ ( ( F ` x ) =/= Z /\ ph ) ) ) )
5	4	rexbidv2 3048	1 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( E. x e. ( F supp Z ) ph <-> E. x e. X ( ( F ` x ) =/= Z /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   E.wrex 2913   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  mdegldg  23826