Theorem fsuppeqg 6447

Description: Version of fsuppeq 6446 avoiding ax-coll 4224 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 5507	. . . . 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 527	. . . 4 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> F e. V )
4		simplr 529	. . . 4 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> Z e. W )
5		suppimacnvfn 6445	. . . 4 |- ( ( F Fn I /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
6	2, 3, 4, 5	syl3anc 1274	. . 3 |- ( ( ( F e. V /\ Z e. W ) /\ F : I --> S ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
7		ffun 5510	. . . . . . 7 |- ( F : I --> S -> Fun F )
8		inpreima 5802	. . . . . . 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 5124	. . . . . . . 8 |- ( `' F " ( _V \ { Z } ) ) C_ dom F
11		fdm 5513	. . . . . . . . 9 |- ( F : I --> S -> dom F = I )
12		fimacnv 5805	. . . . . . . . 9 |- ( F : I --> S -> ( `' F " S ) = I )
13	11, 12	eqtr4d 2268	. . . . . . . 8 |- ( F : I --> S -> dom F = ( `' F " S ) )
14	10, 13	sseqtrid 3287	. . . . . . 7 |- ( F : I --> S -> ( `' F " ( _V \ { Z } ) ) C_ ( `' F " S ) )
15		sseqin2 3439	. . . . . . 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 2265	. . . . 5 |- ( F : I --> S -> ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( _V \ { Z } ) ) )
18		invdif 3462	. . . . . 6 |- ( S i^i ( _V \ { Z } ) ) = ( S \ { Z } )
19	18	imaeq2i 5098	. . . . 5 |- ( `' F " ( S i^i ( _V \ { Z } ) ) ) = ( `' F " ( S \ { Z } ) )
20	17, 19	eqtr3di 2280	. . . 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 2265	. 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 1398   e. wcel 2203   _Vcvv 2812   \ cdif 3207   i^i cin 3209   C_ wss 3210   {csn 3688   `'ccnv 4747   dom cdm 4748   "cima 4751   Fun wfun 5345   Fn wfn 5346   -->wf 5347  (class class class)co 6049   supp csupp 6434

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-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

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-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  fcdmnn0suppg  9549