Theorem suppun2 32765

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 6101	. . . 4 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
2	1	imaeq1i 6017	. . 3 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
3		imaundir 6109	. . 3 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2760	. 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 7701	. . 3 ⊢ (𝜑 → (𝐹 ∪ 𝐺) ∈ V)
8		suppun2.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
9		suppimacnv 8118	. . 3 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ 𝑋) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
10	7, 8, 9	syl2anc 585	. 2 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
11		suppimacnv 8118	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑋) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	5, 8, 11	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13		suppimacnv 8118	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑍 ∈ 𝑋) → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
14	6, 8, 13	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐺 supp 𝑍) = (◡𝐺 “ (V ∖ {𝑍})))
15	12, 14	uneq12d 4122	. 2 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍}))))
16	4, 10, 15	3eqtr4a 2798	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900  {csn 4581  ^◡ccnv 5624   “ cima 5628  (class class class)co 7360   supp csupp 8104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-supp 8105

This theorem is referenced by:  elrgspnlem4  33329  extvfvcl  33703  esplyind  33733