Theorem elsuppfng 6455

Description: An element of the support of a function with a given domain. This version of elsuppfn 6456 assumes F is a set rather than its domain X, avoiding ax-coll 4230. (Contributed by SN, 5-Aug-2024.)

Assertion

Ref	Expression
elsuppfng	|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )

Proof of Theorem elsuppfng

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppvalfng 6453	. . 3 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
2	1	eleq2d 2304	. 2 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> S e. { i e. X | ( F ` i ) =/= Z } ) )
3		fveq2 5675	. . . 4 |- ( i = S -> ( F ` i ) = ( F ` S ) )
4	3	neeq1d 2432	. . 3 |- ( i = S -> ( ( F ` i ) =/= Z <-> ( F ` S ) =/= Z ) )
5	4	elrab 2976	. 2 |- ( S e. { i e. X | ( F ` i ) =/= Z } <-> ( S e. X /\ ( F ` S ) =/= Z ) )
6	2, 5	bitrdi 196	1 |- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   supp csupp 6448

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  suppimacnvfn  6459  fczsupp0  6472  suppssdc  6473  suppssrgst  6475