Theorem suppun 8210

Description: The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)

Hypothesis

Ref	Expression
suppun.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)

Assertion

Ref	Expression
suppun	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))

Proof of Theorem suppun

Step	Hyp	Ref	Expression
1		ssun1 4177	. . . . . 6 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
2		cnvun 6161	. . . . . . . 8 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
3	2	imaeq1i 6074	. . . . . . 7 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
4		imaundir 6169	. . . . . . 7 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
5	3, 4	eqtri 2764	. . . . . 6 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
6	1, 5	sseqtrri 4032	. . . . 5 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍}))
7	6	a1i 11	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
8		suppimacnv 8200	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
9	8	adantr 480	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
10		suppun.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
11		unexg 7764	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
12	11	adantlr 715	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
13	10, 12	sylan2 593	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 ∪ 𝐺) ∈ V)
14		simplr 768	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15		suppimacnv 8200	. . . . 5 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
16	13, 14, 15	syl2anc 584	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
17	7, 9, 16	3sstr4d 4038	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
18	17	ex 412	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)))
19		supp0prc 8189	. . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20		0ss 4399	. . . 4 ⊢ ∅ ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)
21	19, 20	eqsstrdi 4027	. . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
22	21	a1d 25	. 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)))
23	18, 22	pm2.61i 182	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ∖ cdif 3947   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  {csn 4625  ^◡ccnv 5683   “ cima 5687  (class class class)co 7432   supp csupp 8186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-supp 8187

This theorem is referenced by:  fsuppunbi  9430  gsumzaddlem  19940