Theorem suppofssd 8143

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 7391	. . 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 4177	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	1, 2, 3, 4, 4, 5	off 7638	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺):𝐴⟶V)
7		eldif 3909	. . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))))
8		ioran 985	. . . . . 6 ⊢ (¬ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
9		elun 4103	. . . . . 6 ⊢ (𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)))
10	8, 9	xchnxbir 333	. . . . 5 ⊢ (¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
11	10	anbi2i 623	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
12	7, 11	bitri 275	. . 3 ⊢ (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
13	2	ffnd 6661	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		suppofssd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
15		elsuppfn 8110	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
16	13, 4, 14, 15	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
17	16	notbid 318	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
18	17	biimpd 229	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
19	3	ffnd 6661	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
20		elsuppfn 8110	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
21	19, 4, 14, 20	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
22	21	notbid 318	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) ↔ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
23	22	biimpd 229	. . . . . . 7 ⊢ (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) → ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
24	18, 23	anim12d 609	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))))
25	24	anim2d 612	. . . . 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 406	. . . 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		pm3.2 469	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐹‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍)))
28	27	necon1bd 2948	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) → (𝐹‘𝑘) = 𝑍))
29		pm3.2 469	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → ((𝐺‘𝑘) ≠ 𝑍 → (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))
30	29	necon1bd 2948	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍) → (𝐺‘𝑘) = 𝑍))
31	28, 30	anim12d 609	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)) → ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
32	31	imdistani 568	. . . . 5 ⊢ ((𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))) → (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍)))
33	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐹 Fn 𝐴)
34	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐺 Fn 𝐴)
35	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝐴 ∈ 𝑉)
36		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝑘 ∈ 𝐴)
37		fnfvof 7637	. . . . . . 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 838	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = ((𝐹‘𝑘)𝑋(𝐺‘𝑘)))
39		oveq12 7365	. . . . . . 7 ⊢ (((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍) → ((𝐹‘𝑘)𝑋(𝐺‘𝑘)) = (𝑍𝑋𝑍))
40	39	ad2antll 729	. . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
43	38, 40, 42	3eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
44	32, 43	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
45	26, 44	syldan 591	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
46	12, 45	sylan2b 594	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
47	6, 46	suppss 8134	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∘_f cof 7618   supp csupp 8100

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-supp 8101

This theorem is referenced by:  psrbagaddcl  21878  mhpmulcl  22090  mhpaddcl  22092  naddcnff  43546