Theorem funsssuppss 6457

Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)

Assertion

Ref	Expression
funsssuppss	|- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )

Proof of Theorem funsssuppss

Dummy variables x f i z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6435	. . . . . 6 |- supp = ( f e. _V , z e. _V |-> { i e. dom f | ( f " { i } ) =/= { z } } )
2	1	elmpocl2 6250	. . . . 5 |- ( x e. ( F supp Z ) -> Z e. _V )
3		funss 5370	. . . . . . . . . . . 12 |- ( F C_ G -> ( Fun G -> Fun F ) )
4	3	impcom 125	. . . . . . . . . . 11 |- ( ( Fun G /\ F C_ G ) -> Fun F )
5	4	funfnd 5382	. . . . . . . . . 10 |- ( ( Fun G /\ F C_ G ) -> F Fn dom F )
6		funfn 5381	. . . . . . . . . . . 12 |- ( Fun G <-> G Fn dom G )
7	6	biimpi 120	. . . . . . . . . . 11 |- ( Fun G -> G Fn dom G )
8	7	adantr 276	. . . . . . . . . 10 |- ( ( Fun G /\ F C_ G ) -> G Fn dom G )
9	5, 8	jca 306	. . . . . . . . 9 |- ( ( Fun G /\ F C_ G ) -> ( F Fn dom F /\ G Fn dom G ) )
10	9	3adant3 1044	. . . . . . . 8 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F Fn dom F /\ G Fn dom G ) )
11	10	adantr 276	. . . . . . 7 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F Fn dom F /\ G Fn dom G ) )
12		dmss 4954	. . . . . . . . . 10 |- ( F C_ G -> dom F C_ dom G )
13	12	3ad2ant2 1046	. . . . . . . . 9 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> dom F C_ dom G )
14	13	adantr 276	. . . . . . . 8 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom F C_ dom G )
15		dmexg 5020	. . . . . . . . . 10 |- ( G e. V -> dom G e. _V )
16	15	3ad2ant3 1047	. . . . . . . . 9 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> dom G e. _V )
17	16	adantr 276	. . . . . . . 8 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> dom G e. _V )
18		simpr 110	. . . . . . . 8 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> Z e. _V )
19	14, 17, 18	3jca 1204	. . . . . . 7 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) )
20	11, 19	jca 306	. . . . . 6 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( ( F Fn dom F /\ G Fn dom G ) /\ ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) ) )
21		funssfv 5695	. . . . . . . . . . 11 |- ( ( Fun G /\ F C_ G /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
22	21	3expa 1230	. . . . . . . . . 10 |- ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
23		eqeq1 2239	. . . . . . . . . . 11 |- ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z <-> ( F ` x ) = Z ) )
24	23	biimpd 144	. . . . . . . . . 10 |- ( ( G ` x ) = ( F ` x ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
25	22, 24	syl 14	. . . . . . . . 9 |- ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
26	25	ralrimiva 2615	. . . . . . . 8 |- ( ( Fun G /\ F C_ G ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
27	26	3adant3 1044	. . . . . . 7 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
28	27	adantr 276	. . . . . 6 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
29		suppfnss 6456	. . . . . 6 |- ( ( ( F Fn dom F /\ G Fn dom G ) /\ ( dom F C_ dom G /\ dom G e. _V /\ Z e. _V ) ) -> ( A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )
30	20, 28, 29	sylc 62	. . . . 5 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ Z e. _V ) -> ( F supp Z ) C_ ( G supp Z ) )
31	2, 30	sylan2 286	. . . 4 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ x e. ( F supp Z ) ) -> ( F supp Z ) C_ ( G supp Z ) )
32		simpr 110	. . . 4 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ x e. ( F supp Z ) ) -> x e. ( F supp Z ) )
33	31, 32	sseldd 3238	. . 3 |- ( ( ( Fun G /\ F C_ G /\ G e. V ) /\ x e. ( F supp Z ) ) -> x e. ( G supp Z ) )
34	33	ex 115	. 2 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( x e. ( F supp Z ) -> x e. ( G supp Z ) ) )
35	34	ssrdv 3243	1 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   {crab 2524   _Vcvv 2812   C_ wss 3210   {csn 3688   dom cdm 4748   "cima 4751   Fun wfun 5345   Fn wfn 5346   ` cfv 5351  (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-coll 4224  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-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  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-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  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-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by: (None)