Theorem ressuppss 6467

Description: The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)

Assertion

Ref	Expression
ressuppss	|- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )

Proof of Theorem ressuppss

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel2 3410	. . . . . . . 8 |- ( b e. ( B i^i dom F ) -> b e. dom F )
2		dmres 5064	. . . . . . . 8 |- dom ( F |` B ) = ( B i^i dom F )
3	1, 2	eleq2s 2329	. . . . . . 7 |- ( b e. dom ( F |` B ) -> b e. dom F )
4	3	ad2antrl 490	. . . . . 6 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> b e. dom F )
5		elinel1 3409	. . . . . . . . 9 |- ( b e. ( B i^i dom F ) -> b e. B )
6	5, 2	eleq2s 2329	. . . . . . . 8 |- ( b e. dom ( F |` B ) -> b e. B )
7	6	ad2antrl 490	. . . . . . 7 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> b e. B )
8		snssi 3843	. . . . . . . . . . . 12 |- ( b e. B -> { b } C_ B )
9		resima2 5077	. . . . . . . . . . . 12 |- ( { b } C_ B -> ( ( F |` B ) " { b } ) = ( F " { b } ) )
10	8, 9	syl 14	. . . . . . . . . . 11 |- ( b e. B -> ( ( F |` B ) " { b } ) = ( F " { b } ) )
11	10	neeq1d 2432	. . . . . . . . . 10 |- ( b e. B -> ( ( ( F |` B ) " { b } ) =/= { Z } <-> ( F " { b } ) =/= { Z } ) )
12	11	biimpd 144	. . . . . . . . 9 |- ( b e. B -> ( ( ( F |` B ) " { b } ) =/= { Z } -> ( F " { b } ) =/= { Z } ) )
13	12	adantld 278	. . . . . . . 8 |- ( b e. B -> ( ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) -> ( F " { b } ) =/= { Z } ) )
14	13	adantld 278	. . . . . . 7 |- ( b e. B -> ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( F " { b } ) =/= { Z } ) )
15	7, 14	mpcom 36	. . . . . 6 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( F " { b } ) =/= { Z } )
16	4, 15	jca 306	. . . . 5 |- ( ( ( F e. V /\ Z e. W ) /\ ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) ) -> ( b e. dom F /\ ( F " { b } ) =/= { Z } ) )
17	16	ex 115	. . . 4 |- ( ( F e. V /\ Z e. W ) -> ( ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) -> ( b e. dom F /\ ( F " { b } ) =/= { Z } ) ) )
18	17	ss2abdv 3315	. . 3 |- ( ( F e. V /\ Z e. W ) -> { b | ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) } C_ { b | ( b e. dom F /\ ( F " { b } ) =/= { Z } ) } )
19		df-rab 2531	. . 3 |- { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } = { b | ( b e. dom ( F |` B ) /\ ( ( F |` B ) " { b } ) =/= { Z } ) }
20		df-rab 2531	. . 3 |- { b e. dom F | ( F " { b } ) =/= { Z } } = { b | ( b e. dom F /\ ( F " { b } ) =/= { Z } ) }
21	18, 19, 20	3sstr4g 3285	. 2 |- ( ( F e. V /\ Z e. W ) -> { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } C_ { b e. dom F | ( F " { b } ) =/= { Z } } )
22		resexg 5083	. . 3 |- ( F e. V -> ( F |` B ) e. _V )
23		suppval 6450	. . 3 |- ( ( ( F |` B ) e. _V /\ Z e. W ) -> ( ( F |` B ) supp Z ) = { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } )
24	22, 23	sylan 283	. 2 |- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) = { b e. dom ( F |` B ) | ( ( F |` B ) " { b } ) =/= { Z } } )
25		suppval 6450	. 2 |- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) = { b e. dom F | ( F " { b } ) =/= { Z } } )
26	21, 24, 25	3sstr4d 3287	1 |- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   =/= wne 2414   {crab 2526   _Vcvv 2815   i^i cin 3213   C_ wss 3214   {csn 3694   dom cdm 4754   |` cres 4756   "cima 4757  (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-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by: (None)