Theorem suppvalfn 6475

Description: The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)

Assertion

Ref	Expression
suppvalfn	|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )

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

Proof of Theorem suppvalfn

Step	Hyp	Ref	Expression
1		fnfun 5476	. . . 4 |- ( F Fn X -> Fun F )
2	1	3ad2ant1 1049	. . 3 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> Fun F )
3		fnex 5931	. . . 4 |- ( ( F Fn X /\ X e. V ) -> F e. _V )
4	3	3adant3 1048	. . 3 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> F e. _V )
5		simp3 1030	. . 3 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> Z e. W )
6		suppval1 6473	. . 3 |- ( ( Fun F /\ F e. _V /\ Z e. W ) -> ( F supp Z ) = { i e. dom F | ( F ` i ) =/= Z } )
7	2, 4, 5, 6	syl3anc 1278	. 2 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( F supp Z ) = { i e. dom F | ( F ` i ) =/= Z } )
8		fndm 5478	. . . 4 |- ( F Fn X -> dom F = X )
9	8	3ad2ant1 1049	. . 3 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> dom F = X )
10	9	rabeqdv 2815	. 2 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> { i e. dom F | ( F ` i ) =/= Z } = { i e. X | ( F ` i ) =/= Z } )
11	7, 10	eqtrd 2271	1 |- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   dom cdm 4772   Fun wfun 5369   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   supp csupp 6469

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  elsuppfn  6477  rrgsupp  14557