Theorem suppun2 32698

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 6164	. . . 4 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
2	1	imaeq1i 6076	. . 3 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
3		imaundir 6172	. . 3 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2762	. 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 7772	. . 3 ⊢ (𝜑 → (𝐹 ∪ 𝐺) ∈ V)
8		suppun2.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
9		suppimacnv 8197	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑋) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
10	7, 8, 9	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
11		suppimacnv 8197	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑋) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	5, 8, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13		suppimacnv 8197	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑍 ∈ 𝑋) → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
14	6, 8, 13	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
15	12, 14	uneq12d 4178	. 2 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍}))))
16	4, 10, 15	3eqtr4a 2800	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  Vcvv 3477   ∖ cdif 3959   ∪ cun 3960  {csn 4630  ^◡ccnv 5687   “ cima 5691  (class class class)co 7430   supp csupp 8183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184

This theorem is referenced by:  elrgspnlem4  33234