Theorem suppofssd 8003

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 7303	. . 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 4157	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	1, 2, 3, 4, 4, 5	off 7542	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑋𝐺):𝐴⟶V)
7		eldif 3901	. . . 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 4087	. . . . . 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 622	. . . 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 6597	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14		suppofssd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
15		elsuppfn 7971	. . . . . . . . . 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 6597	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
20		elsuppfn 7971	. . . . . . . . . 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 608	. . . . . 6 ⊢ (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍))))
25	24	anim2d 611	. . . . 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 2962	. . . . . . 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 2962	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍) → (𝐺‘𝑘) = 𝑍))
31	28, 30	anim12d 608	. . . . . 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 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → 𝑘 ∈ 𝐴)
37		fnfvof 7541	. . . . . . 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 7277	. . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
43	38, 40, 42	3eqtrd 2783	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ((𝐹‘𝑘) = 𝑍 ∧ (𝐺‘𝑘) = 𝑍))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
44	32, 43	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ (𝑘 ∈ 𝐴 ∧ (𝐹‘𝑘) ≠ 𝑍) ∧ ¬ (𝑘 ∈ 𝐴 ∧ (𝐺‘𝑘) ≠ 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
45	26, 44	syldan 590	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
46	12, 45	sylan2b 593	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹 ∘f 𝑋𝐺)‘𝑘) = 𝑍)
47	6, 46	suppss 7994	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  Vcvv 3430   ∖ cdif 3888   ∪ cun 3889   ⊆ wss 3891   Fn wfn 6425  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268   ∘_f cof 7522   supp csupp 7961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-supp 7962

This theorem is referenced by:  psrbagaddcl  21112  mhpmulcl  21320  mhpaddcl  21322