Theorem rexsupp 6497

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 6487	. . . 4 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( x e. ( F supp Z ) <-> ( x e. X /\ ( F ` x ) =/= Z ) ) )
2	1	anbi1d 469	. . 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 405	. . 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 2553	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   =/= wne 2420   E.wrex 2529   Fn wfn 5375   ` cfv 5380  (class class class)co 6088   supp csupp 6479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480

This theorem is used by: (None)