Theorem suppfnss 8214

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 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (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 1213	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐴 ⊆ 𝐵)
2		fndm 6647	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐹 = 𝐴)
4		fndm 6647	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
5	4	ad2antlr 740	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐺 = 𝐵)
6	1, 3, 5	3sstr4d 3964	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐹 ⊆ dom 𝐺)
7	6	adantr 486	. . . 4 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → dom 𝐹 ⊆ dom 𝐺)
8	2	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
9	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
10		fveqeq2 6899	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐺‘𝑥) = 𝑍 ↔ (𝐺‘𝑦) = 𝑍))
11		fveqeq2 6899	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑦) = 𝑍))
12	10, 11	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) ↔ ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍)))
13	12	rspcv 3572	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍)))
14	9, 13	biimtrdi 256	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝑦 ∈ dom 𝐹 → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍))))
15	14	com23 87	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝑦 ∈ dom 𝐹 → ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍))))
16	15	imp31 423	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍))
17	16	necon3d 2976	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ≠ 𝑍 → (𝐺‘𝑦) ≠ 𝑍))
18	17	ex 418	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → (𝑦 ∈ dom 𝐹 → ((𝐹‘𝑦) ≠ 𝑍 → (𝐺‘𝑦) ≠ 𝑍)))
19	18	com23 87	. . . . 5 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → ((𝐹‘𝑦) ≠ 𝑍 → (𝑦 ∈ dom 𝐹 → (𝐺‘𝑦) ≠ 𝑍)))
20	19	3imp 1128	. . . 4 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) ∧ (𝐹‘𝑦) ≠ 𝑍 ∧ 𝑦 ∈ dom 𝐹) → (𝐺‘𝑦) ≠ 𝑍)
21	7, 20	rabssrabd 4007	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍})
22		fnfun 6644	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
23	22	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → Fun 𝐹)
24		simpl 488	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
25		ssexg 5281	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
26	25	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐴 ∈ V)
27		fnex 7227	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
28	24, 26, 27	syl2an 608	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐹 ∈ V)
29		simpr3 1215	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝑍 ∈ 𝑊)
30		suppval1 8191	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍})
31	23, 28, 29, 30	syl3anc 1398	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍})
32		fnfun 6644	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
33	32	ad2antlr 740	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → Fun 𝐺)
34		simpr 490	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
35		simp2 1155	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐵 ∈ 𝑉)
36		fnex 7227	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
37	34, 35, 36	syl2an 608	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐺 ∈ V)
38		suppval1 8191	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍})
39	33, 37, 29, 38	syl3anc 1398	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍})
40	31, 39	sseq12d 3942	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍}))
41	40	adantr 486	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍} ⊆ {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍}))
42	21, 41	mpbird 260	. 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) ∧ ∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
43	42	ex 418	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3896  dom cdm 5651  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  (class class class)co 7424   supp csupp 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-supp 8185

This theorem is used by:  funsssuppss  8215  suppofss1d  8229  suppofss2d  8230  lincresunit2  49726