Theorem suppfnss 6470

Description: The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)

Assertion

Ref	Expression
suppfnss	|- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )

Distinct variable groups:   x, A   x, F   x, G   x, Z

Allowed substitution hints:   B( x)   V( x)   W( x)

Proof of Theorem suppfnss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1030	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> A C_ B )
2		fndm 5460	. . . . . . 7 |- ( F Fn A -> dom F = A )
3	2	ad2antrr 488	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom F = A )
4		fndm 5460	. . . . . . 7 |- ( G Fn B -> dom G = B )
5	4	ad2antlr 489	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom G = B )
6	1, 3, 5	3sstr4d 3287	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom F C_ dom G )
7	6	adantr 276	. . . 4 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> dom F C_ dom G )
8	2	eleq2d 2304	. . . . . . . . . . . 12 |- ( F Fn A -> ( y e. dom F <-> y e. A ) )
9	8	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( y e. dom F <-> y e. A ) )
10		fveqeq2 5684	. . . . . . . . . . . . 13 |- ( x = y -> ( ( G ` x ) = Z <-> ( G ` y ) = Z ) )
11		fveqeq2 5684	. . . . . . . . . . . . 13 |- ( x = y -> ( ( F ` x ) = Z <-> ( F ` y ) = Z ) )
12	10, 11	imbi12d 234	. . . . . . . . . . . 12 |- ( x = y -> ( ( ( G ` x ) = Z -> ( F ` x ) = Z ) <-> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) )
13	12	rspcv 2919	. . . . . . . . . . 11 |- ( y e. A -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) )
14	9, 13	biimtrdi 163	. . . . . . . . . 10 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( y e. dom F -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) ) )
15	14	com23 78	. . . . . . . . 9 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( y e. dom F -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) ) )
16	15	imp31 256	. . . . . . . 8 |- ( ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) /\ y e. dom F ) -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) )
17	16	necon3d 2458	. . . . . . 7 |- ( ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) /\ y e. dom F ) -> ( ( F ` y ) =/= Z -> ( G ` y ) =/= Z ) )
18	17	ex 115	. . . . . 6 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> ( y e. dom F -> ( ( F ` y ) =/= Z -> ( G ` y ) =/= Z ) ) )
19	18	com23 78	. . . . 5 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> ( ( F ` y ) =/= Z -> ( y e. dom F -> ( G ` y ) =/= Z ) ) )
20	19	3imp 1220	. . . 4 |- ( ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) /\ ( F ` y ) =/= Z /\ y e. dom F ) -> ( G ` y ) =/= Z )
21	7, 20	rabssrabd 3329	. . 3 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> { y e. dom F | ( F ` y ) =/= Z } C_ { y e. dom G | ( G ` y ) =/= Z } )
22		fnfun 5458	. . . . . . 7 |- ( F Fn A -> Fun F )
23	22	ad2antrr 488	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Fun F )
24		simpl 109	. . . . . . 7 |- ( ( F Fn A /\ G Fn B ) -> F Fn A )
25		ssexg 4254	. . . . . . . 8 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
26	25	3adant3 1044	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> A e. _V )
27		fnex 5911	. . . . . . 7 |- ( ( F Fn A /\ A e. _V ) -> F e. _V )
28	24, 26, 27	syl2an 289	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> F e. _V )
29		simpr3 1032	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Z e. W )
30		suppval1 6452	. . . . . 6 |- ( ( Fun F /\ F e. _V /\ Z e. W ) -> ( F supp Z ) = { y e. dom F | ( F ` y ) =/= Z } )
31	23, 28, 29, 30	syl3anc 1274	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( F supp Z ) = { y e. dom F | ( F ` y ) =/= Z } )
32		fnfun 5458	. . . . . . 7 |- ( G Fn B -> Fun G )
33	32	ad2antlr 489	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Fun G )
34		simpr 110	. . . . . . 7 |- ( ( F Fn A /\ G Fn B ) -> G Fn B )
35		simp2 1025	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> B e. V )
36		fnex 5911	. . . . . . 7 |- ( ( G Fn B /\ B e. V ) -> G e. _V )
37	34, 35, 36	syl2an 289	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> G e. _V )
38		suppval1 6452	. . . . . 6 |- ( ( Fun G /\ G e. _V /\ Z e. W ) -> ( G supp Z ) = { y e. dom G | ( G ` y ) =/= Z } )
39	33, 37, 29, 38	syl3anc 1274	. . . . 5 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( G supp Z ) = { y e. dom G | ( G ` y ) =/= Z } )
40	31, 39	sseq12d 3273	. . . 4 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( ( F supp Z ) C_ ( G supp Z ) <-> { y e. dom F | ( F ` y ) =/= Z } C_ { y e. dom G | ( G ` y ) =/= Z } ) )
41	40	adantr 276	. . 3 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> ( ( F supp Z ) C_ ( G supp Z ) <-> { y e. dom F | ( F ` y ) =/= Z } C_ { y e. dom G | ( G ` y ) =/= Z } ) )
42	21, 41	mpbird 167	. 2 |- ( ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) /\ A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) ) -> ( F supp Z ) C_ ( G supp Z ) )
43	42	ex 115	1 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   {crab 2526   _Vcvv 2815   C_ wss 3214   dom cdm 4754   Fun wfun 5351   Fn wfn 5352   ` cfv 5357  (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-coll 4230  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-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  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-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  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-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  funsssuppss  6471  suppofss1dcl  6477  suppofss2dcl  6478