Theorem suppun2 33104

Description: The support of a union is the union of the supports. (Contributed by Thierry Arnoux, 5-Oct-2025.)

Hypotheses

Ref	Expression
suppun2.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
suppun2.2	⊢ (𝜑 → 𝐺 ∈ 𝑊)
suppun2.3	⊢ (𝜑 → 𝑍 ∈ 𝑋)

Assertion

Ref	Expression
suppun2	⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Proof of Theorem suppun2

Step	Hyp	Ref	Expression
1		cnvun 6141	. . . 4 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
2	1	imaeq1i 6061	. . 3 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
3		imaundir 6150	. . 3 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2788	. 2 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
5		suppun2.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
6		suppun2.2	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
7	5, 6	unexd 7759	. . 3 ⊢ (𝜑 → (𝐹 ∪ 𝐺) ∈ V)
8		suppun2.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
9		suppimacnv 8176	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑋) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
10	7, 8, 9	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
11		suppimacnv 8176	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑋) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	5, 8, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13		suppimacnv 8176	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑍 ∈ 𝑋) → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
14	6, 8, 13	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
15	12, 14	uneq12d 4123	. 2 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍}))))
16	4, 10, 15	3eqtr4a 2826	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904  {csn 4591  ^◡ccnv 5662   “ cima 5666  (class class class)co 7419   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-sep 5259  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-rab 3419  df-v 3459  df-sbc 3747  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-br 5112  df-opab 5176  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-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-supp 8163

This theorem is used by:  elrgspnlem4  33633  extvfvcl  33994  esplyind  34033