Theorem suppfnss 6491

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 1034	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> A C_ B )
2		fndm 5478	. . . . . . 7 |- ( F Fn A -> dom F = A )
3	2	ad2antrr 492	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom F = A )
4		fndm 5478	. . . . . . 7 |- ( G Fn B -> dom G = B )
5	4	ad2antlr 493	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> dom G = B )
6	1, 3, 5	3sstr4d 3293	. . . . 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 2308	. . . . . . . . . . . 12 |- ( F Fn A -> ( y e. dom F <-> y e. A ) )
9	8	ad2antrr 492	. . . . . . . . . . 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 5702	. . . . . . . . . . . . 13 |- ( x = y -> ( ( G ` x ) = Z <-> ( G ` y ) = Z ) )
11		fveqeq2 5702	. . . . . . . . . . . . 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 2925	. . . . . . . . . . 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 2464	. . . . . . 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 1224	. . . 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 3335	. . 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 5476	. . . . . . 7 |- ( F Fn A -> Fun F )
23	22	ad2antrr 492	. . . . . 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 4270	. . . . . . . 8 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
26	25	3adant3 1048	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> A e. _V )
27		fnex 5931	. . . . . . 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 1036	. . . . . 6 |- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> Z e. W )
30		suppval1 6473	. . . . . 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 1278	. . . . 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 5476	. . . . . . 7 |- ( G Fn B -> Fun G )
33	32	ad2antlr 493	. . . . . 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 1029	. . . . . . 7 |- ( ( A C_ B /\ B e. V /\ Z e. W ) -> B e. V )
36		fnex 5931	. . . . . . 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 6473	. . . . . 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 1278	. . . . 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 3279	. . . 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {crab 2532   _Vcvv 2821   C_ wss 3220   dom cdm 4772   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:  funsssuppss  6492  suppofss1dcl  6498  suppofss2dcl  6499