Theorem fsuppeqg 6482

Description: Version of fsuppeq 6481 avoiding ax-coll 4244 by assuming F is a set rather than its domain I. (Contributed by SN, 30-Jul-2024.)

Assertion

Ref	Expression
fsuppeqg	|- ( ( F e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )

Proof of Theorem fsuppeqg

Step	Hyp	Ref	Expression
1		ffn 5531	. . . . 5 |- ( F : I --> S -> F Fn I )
2	1	adantl 277	. . . 4 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> F Fn I )
3		simpll 531	. . . 4 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> F e. V )
4		simplr 533	. . . 4 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> Z e. W )
5		suppimacnvfn 6480	. . . 4 |- ( ( F Fn I /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
6	2, 3, 4, 5	syl3anc 1278	. . 3 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
7		ffun 5534	. . . . . . 7 |- ( F : I --> S -> Fun F )
8		inpreima 5828	. . . . . . 7 |- ( Fun F -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
9	7, 8	syl 14	. . . . . 6 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) )
10		cnvimass 5148	. . . . . . . 8 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
11		fdm 5537	. . . . . . . . 9 |- ( F : I --> S -> dom F = I )
12		fimacnv 5831	. . . . . . . . 9 |- ( F : I --> S -> ( `' F " S ) = I )
13	11, 12	eqtr4d 2274	. . . . . . . 8 |- ( F : I --> S -> dom F = ( `' F " S ) )
14	10, 13	sseqtrid 3298	. . . . . . 7 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) )
15		sseqin2 3450	. . . . . . 7 |- ( ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) <-> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
16	14, 15	sylib 122	. . . . . 6 |- ( F : I --> S -> ( ( `' F " S ) i^i ( `' F " ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
17	9, 16	eqtrd 2271	. . . . 5 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
18		invdif 3473	. . . . . 6 |- ( S i^i ( _V \ { Z } ) ) = ( S \ { Z } )
19	18	imaeq2i 5122	. . . . 5 |- ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( S \ { Z } ) )
20	17, 19	eqtr3di 2286	. . . 4 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
21	20	adantl 277	. . 3 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> ( `' F " ( _V \ { Z } ) ) = ( `' F " ( S \ { Z } ) ) )
22	6, 21	eqtrd 2271	. 2 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) )
23	22	ex 115	1 |- ( ( F e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ cdif 3217   i^i cin 3219   C_ wss 3220   {csn 3708   `'ccnv 4771   dom cdm 4772   "cima 4775   Fun wfun 5369   Fn wfn 5370   -->wf 5371  (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-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-rab 2537  df-v 2823  df-sbc 3052  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-br 4129  df-opab 4191  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-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  fcdmnn0suppg  9600