Theorem suppfnss 8119

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 1195	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐴 ⊆ 𝐵)
2		fndm 6584	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐹 = 𝐴)
4		fndm 6584	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
5	4	ad2antlr 727	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐺 = 𝐵)
6	1, 3, 5	3sstr4d 3985	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → dom 𝐹 ⊆ dom 𝐺)
7	6	adantr 480	. . . 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 2817	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
9	8	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
10		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐺‘𝑥) = 𝑍 ↔ (𝐺‘𝑦) = 𝑍))
11		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑦) = 𝑍))
12	10, 11	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) ↔ ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍)))
13	12	rspcv 3568	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → ((𝐺‘𝑦) = 𝑍 → (𝐹‘𝑦) = 𝑍)))
14	9, 13	biimtrdi 253	. . . . . . . . . 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 86	. . . . . . . . 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 417	. . . . . . . 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 2949	. . . . . . 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 412	. . . . . 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 86	. . . . 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 1110	. . . 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 4030	. . 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 6581	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
23	22	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → Fun 𝐹)
24		simpl 482	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐹 Fn 𝐴)
25		ssexg 5259	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
26	25	3adant3 1132	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐴 ∈ V)
27		fnex 7151	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐹 ∈ V)
28	24, 26, 27	syl2an 596	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐹 ∈ V)
29		simpr3 1197	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝑍 ∈ 𝑊)
30		suppval1 8096	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍})
31	23, 28, 29, 30	syl3anc 1373	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (𝐹 supp 𝑍) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ≠ 𝑍})
32		fnfun 6581	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
33	32	ad2antlr 727	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → Fun 𝐺)
34		simpr 484	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐺 Fn 𝐵)
35		simp2 1137	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → 𝐵 ∈ 𝑉)
36		fnex 7151	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
37	34, 35, 36	syl2an 596	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → 𝐺 ∈ V)
38		suppval1 8096	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐺 supp 𝑍) = {𝑦 ∈ dom 𝐺 ∣ (𝐺‘𝑦) ≠ 𝑍})
39	33, 37, 29, 38	syl3anc 1373	. . . . 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 3963	. . . 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 480	. . 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 257	. 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 412	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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395  Vcvv 3436   ⊆ wss 3897  dom cdm 5614  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   supp csupp 8090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-supp 8091

This theorem is referenced by:  funsssuppss  8120  suppofss1d  8134  suppofss2d  8135  lincresunit2  48589