Theorem funsssuppss 6492

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 6470	. . . . . 6 |- supp = ( f e. _V , z e. _V |-> { i e. dom f | ( f " { i } ) =/= { z } } )
2	1	elmpocl2 6280	. . . . 5 |- ( x e. ( F supp Z ) -> Z e. _V )
3		funss 5394	. . . . . . . . . . . 12 |- ( F C_ G -> ( Fun G -> Fun F ) )
4	3	impcom 125	. . . . . . . . . . 11 |- ( ( Fun G /\ F C_ G ) -> Fun F )
5	4	funfnd 5406	. . . . . . . . . 10 |- ( ( Fun G /\ F C_ G ) -> F Fn dom F )
6		funfn 5405	. . . . . . . . . . . 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 1048	. . . . . . . 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 4978	. . . . . . . . . 10 |- ( F C_ G -> dom F C_ dom G )
13	12	3ad2ant2 1050	. . . . . . . . 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 5044	. . . . . . . . . 10 |- ( G e. V -> dom G e. _V )
16	15	3ad2ant3 1051	. . . . . . . . 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 1208	. . . . . . 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 5719	. . . . . . . . . . 11 |- ( ( Fun G /\ F C_ G /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
22	21	3expa 1234	. . . . . . . . . 10 |- ( ( ( Fun G /\ F C_ G ) /\ x e. dom F ) -> ( G ` x ) = ( F ` x ) )
23		eqeq1 2245	. . . . . . . . . . 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 2623	. . . . . . . 8 |- ( ( Fun G /\ F C_ G ) -> A. x e. dom F ( ( G ` x ) = Z -> ( F ` x ) = Z ) )
27	26	3adant3 1048	. . . . . . 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 6491	. . . . . 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 3249	. . 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 3254	1 |- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   _Vcvv 2821   C_ wss 3220   {csn 3708   dom cdm 4772   "cima 4775   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: (None)