Theorem suppun2 32777

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 6094	. . . 4 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
2	1	imaeq1i 6010	. . 3 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
3		imaundir 6102	. . 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 7698	. . 3 ⊢ (𝜑 → (𝐹 ∪ 𝐺) ∈ V)
8		suppun2.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
9		suppimacnv 8115	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑋) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
10	7, 8, 9	syl2anc 590	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
11		suppimacnv 8115	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑋) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	5, 8, 11	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13		suppimacnv 8115	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑍 ∈ 𝑋) → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
14	6, 8, 13	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
15	12, 14	uneq12d 4100	. 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 1547   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881  {csn 4556  ^◡ccnv 5618   “ cima 5622  (class class class)co 7357   supp csupp 8101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-supp 8102

This theorem is referenced by:  elrgspnlem4  33327  extvfvcl  33729  esplyind  33768