Theorem suppun2 32657

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 6103	. . . 4 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
2	1	imaeq1i 6017	. . 3 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
3		imaundir 6111	. . 3 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
4	2, 3	eqtri 2752	. 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 7710	. . 3 ⊢ (𝜑 → (𝐹 ∪ 𝐺) ∈ V)
8		suppun2.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑋)
9		suppimacnv 8130	. . 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 8130	. . . 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 8130	. . . 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 4128	. 2 ⊢ (𝜑 → ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍}))))
16	4, 10, 15	3eqtr4a 2790	1 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909  {csn 4585  ^◡ccnv 5630   “ cima 5634  (class class class)co 7369   supp csupp 8116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-supp 8117

This theorem is referenced by:  elrgspnlem4  33212