Theorem suppfnss 7320

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.)

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		fndm 5990	. . . . . . . . . . 11 |- ( F Fn A -> dom F = A )
2	1	eleq2d 2687	. . . . . . . . . 10 |- ( F Fn A -> ( y e. dom F <-> y e. A ) )
3	2	ad2antrr 762	. . . . . . . . 9 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( y e. dom F <-> y e. A ) )
4		fveq2 6191	. . . . . . . . . . . 12 |- ( x = y -> ( G ` x ) = ( G ` y ) )
5	4	eqeq1d 2624	. . . . . . . . . . 11 |- ( x = y -> ( ( G ` x ) = Z <-> ( G ` y ) = Z ) )
6		fveq2 6191	. . . . . . . . . . . 12 |- ( x = y -> ( F ` x ) = ( F ` y ) )
7	6	eqeq1d 2624	. . . . . . . . . . 11 |- ( x = y -> ( ( F ` x ) = Z <-> ( F ` y ) = Z ) )
8	5, 7	imbi12d 334	. . . . . . . . . 10 |- ( x = y -> ( ( ( G ` x ) = Z -> ( F ` x ) = Z ) <-> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) )
9	8	rspcv 3305	. . . . . . . . 9 |- ( y e. A -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) )
10	3, 9	syl6bi 243	. . . . . . . 8 |- ( ( ( 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 ) ) ) )
11	10	com23 86	. . . . . . 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 -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) ) ) )
12	11	imp31 448	. . . . . 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 ) -> ( ( G ` y ) = Z -> ( F ` y ) = Z ) )
13	12	necon3d 2815	. . . . 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 ) ) /\ y e. dom F ) -> ( ( F ` y ) =/= Z -> ( G ` y ) =/= Z ) )
14	13	ss2rabdv 3683	. . . 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 ) ) -> { y e. dom F | ( F ` y ) =/= Z } C_ { y e. dom F | ( G ` y ) =/= Z } )
15		simpr1 1067	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> A C_ B )
16	1	ad2antrr 762	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom F = A )
17		fndm 5990	. . . . . . . 8 |- ( G Fn B -> dom G = B )
18	17	ad2antlr 763	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom G = B )
19	15, 16, 18	3sstr4d 3648	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom F C_ dom G )
20	19	adantr 481	. . . . 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 ) ) -> dom F C_ dom G )
21		rabss2 3685	. . . . 5 |- ( dom F C_ dom G -> { y e. dom F | ( G ` y ) =/= Z } C_ { y e. dom G | ( G ` y ) =/= Z } )
22	20, 21	syl 17	. . . 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 ) ) -> { y e. dom F | ( G ` y ) =/= Z } C_ { y e. dom G | ( G ` y ) =/= Z } )
23	14, 22	sstrd 3613	. . 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 } )
24		fnfun 5988	. . . . . . 7 |- ( F Fn A -> Fun F )
25	24	ad2antrr 762	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Fun F )
26		simpl 473	. . . . . . 7 |- ( ( F Fn A /\ G Fn B ) -> F Fn A )
27		ssexg 4804	. . . . . . . 8 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
28	27	3adant3 1081	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> A e. _V )
29		fnex 6481	. . . . . . 7 |- ( ( F Fn A /\ A e. _V ) -> F e. _V )
30	26, 28, 29	syl2an 494	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> F e. _V )
31		simpr3 1069	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Z e. W )
32		suppval1 7301	. . . . . 6 |- ( ( Fun F /\ F e. _V /\ Z e. W ) -> ( F supp Z ) = { y e. dom F | ( F ` y ) =/= Z } )
33	25, 30, 31, 32	syl3anc 1326	. . . . 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 } )
34		fnfun 5988	. . . . . . 7 |- ( G Fn B -> Fun G )
35	34	ad2antlr 763	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Fun G )
36		simpr 477	. . . . . . 7 |- ( ( F Fn A /\ G Fn B ) -> G Fn B )
37		simp2 1062	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> B e. V )
38		fnex 6481	. . . . . . 7 |- ( ( G Fn B /\ B e. V ) -> G e. _V )
39	36, 37, 38	syl2an 494	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> G e. _V )
40		suppval1 7301	. . . . . 6 |- ( ( Fun G /\ G e. _V /\ Z e. W ) -> ( G supp Z ) = { y e. dom G | ( G ` y ) =/= Z } )
41	35, 39, 31, 40	syl3anc 1326	. . . . 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 } )
42	33, 41	sseq12d 3634	. . . 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 } ) )
43	42	adantr 481	. . 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 } ) )
44	23, 43	mpbird 247	. 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 ) )
45	44	ex 450	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  funsssuppss  7321  suppofss1d  7332  suppofss2d  7333  lincresunit2  42267