Theorem suppofssd 8205

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 7454	. . 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 4179	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	1, 2, 3, 4, 4, 5	off 7702	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺):𝐴⟶V)
7		eldif 3916	. . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))))
8		ioran 999	. . . . . 6 ⊢ (¬ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
9		elun 4107	. . . . . 6 ⊢ (𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)))
10	8, 9	xchnxbir 336	. . . . 5 ⊢ (¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
11	10	anbi2i 635	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
12	7, 11	bitri 278	. . 3 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
13	2	ffnd 6710	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		suppofssd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
15		elsuppfn 8172	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
16	13, 4, 14, 15	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
17	16	notbid 321	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
18	17	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
19	3	ffnd 6710	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
20		elsuppfn 8172	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
21	19, 4, 14, 20	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
22	21	notbid 321	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
23	22	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
24	18, 23	anim12d 621	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))))
25	24	anim2d 624	. . . . 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 412	. . . 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		pm3.2 475	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐹‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
28	27	necon1bd 2978	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → (𝐹‘𝑘) = 𝑍))
29		pm3.2 475	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
30	29	necon1bd 2978	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍) → (𝐺‘𝑘) = 𝑍))
31	28, 30	anim12d 621	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)) → ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
32	31	imdistani 579	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))) → (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
33	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐹 Fn 𝐴)
34	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐺 Fn 𝐴)
35	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐴 ∈ 𝑉)
36		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝑘 ∈ 𝐴)
37		fnfvof 7701	. . . . . . 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 852	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = ((𝐹‘𝑘)𝑋(𝐺‘𝑘)))
39		oveq12 7428	. . . . . . 7 ⊢ (((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍) → ((𝐹‘𝑘)𝑋(𝐺‘𝑘)) = (𝑍𝑋𝑍))
40	39	ad2antll 742	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
43	38, 40, 42	3eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
44	32, 43	sylan2 605	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
45	26, 44	syldan 603	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
46	12, 45	sylan2b 606	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
47	6, 46	suppss 8196	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∘_f cof 7682   supp csupp 8162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-supp 8163

This theorem is used by:  psrbagaddcl  22128  mhpmulcl  22366  mhpaddcl  22368  naddcnff  44166