Theorem rexsupp 6453

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 6443	. . . 4 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> ( x e. X /\ ( F ` x ) =/= Z ) ) )
2	1	anbi1d 465	. . 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 401	. . 3 |- ( ( ( x e. X /\ ( F ` x ) =/= Z ) /\ ph ) <-> ( x e. X /\ ( ( F ` x ) =/= Z /\ ph ) ) )
4	2, 3	bitrdi 196	. 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 2545	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   /\ wa 104   <-> wb 105   /\ w3a 1005   e. wcel 2203   =/= wne 2412   E.wrex 2521   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   supp csupp 6435

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-supp 6436

This theorem is referenced by: (None)