Theorem suppofssd 8190

Description: Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.)

Hypotheses

Ref	Expression
suppofssd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppofssd.2	⊢ (𝜑 → 𝑍 ∈ 𝐵)
suppofssd.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
suppofssd.4	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
suppofssd.5	⊢ (𝜑 → (𝑍𝑋𝑍) = 𝑍)

Assertion

Ref	Expression
suppofssd	⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Proof of Theorem suppofssd

Dummy variables 𝑥 𝑦 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7446	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝑋𝑦) ∈ V)
2		suppofssd.3	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3		suppofssd.4	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4		suppofssd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		inidm 4217	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	1, 2, 3, 4, 4, 5	off 7690	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺):𝐴⟶V)
7		eldif 3957	. . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))))
8		ioran 980	. . . . . 6 ⊢ (¬ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
9		elun 4147	. . . . . 6 ⊢ (𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)))
10	8, 9	xchnxbir 332	. . . . 5 ⊢ (¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
11	10	anbi2i 621	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
12	7, 11	bitri 274	. . 3 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
13	2	ffnd 6717	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		suppofssd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
15		elsuppfn 8158	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
16	13, 4, 14, 15	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
17	16	notbid 317	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
18	17	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
19	3	ffnd 6717	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
20		elsuppfn 8158	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
21	19, 4, 14, 20	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
22	21	notbid 317	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
23	22	biimpd 228	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
24	18, 23	anim12d 607	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))))
25	24	anim2d 610	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))) → (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))))
26	25	imp 405	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))))
27		pm3.2 468	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐹‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
28	27	necon1bd 2956	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → (𝐹‘𝑘) = 𝑍))
29		pm3.2 468	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
30	29	necon1bd 2956	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍) → (𝐺‘𝑘) = 𝑍))
31	28, 30	anim12d 607	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)) → ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
32	31	imdistani 567	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))) → (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
33	13	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐹 Fn 𝐴)
34	19	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐺 Fn 𝐴)
35	4	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐴 ∈ 𝑉)
36		simprl 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝑘 ∈ 𝐴)
37		fnfvof 7689	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴)) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = ((𝐹‘𝑘)𝑋(𝐺‘𝑘)))
38	33, 34, 35, 36, 37	syl22anc 835	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = ((𝐹‘𝑘)𝑋(𝐺‘𝑘)))
39		oveq12 7420	. . . . . . 7 ⊢ (((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍) → ((𝐹‘𝑘)𝑋(𝐺‘𝑘)) = (𝑍𝑋𝑍))
40	39	ad2antll 725	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹‘𝑘)𝑋(𝐺‘𝑘)) = (𝑍𝑋𝑍))
41		suppofssd.5	. . . . . . 7 ⊢ (𝜑 → (𝑍𝑋𝑍) = 𝑍)
42	41	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
43	38, 40, 42	3eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
44	32, 43	sylan2 591	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
45	26, 44	syldan 589	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
46	12, 45	sylan2b 592	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
47	6, 46	suppss 8181	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ∘_f cof 7670   supp csupp 8148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-supp 8149

This theorem is referenced by:  psrbagaddcl  21700  mhpmulcl  21911  mhpaddcl  21913  naddcnff  42414