Theorem suppun 8127

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 4119	. . . . . 6 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
2		cnvun 6100	. . . . . . . 8 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
3	2	imaeq1i 6016	. . . . . . 7 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
4		imaundir 6108	. . . . . . 7 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
5	3, 4	eqtri 2760	. . . . . 6 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
6	1, 5	sseqtrri 3972	. . . . 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 8117	. . . . 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 7690	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
12	11	adantlr 716	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
13	10, 12	sylan2 594	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 ∪ 𝐺) ∈ V)
14		simplr 769	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15		suppimacnv 8117	. . . . 5 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
16	13, 14, 15	syl2anc 585	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
17	7, 9, 16	3sstr4d 3978	. . 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 8106	. . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20		0ss 4341	. . . 4 ⊢ ∅ ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)
21	19, 20	eqsstrdi 3967	. . 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 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  ^◡ccnv 5623   “ cima 5627  (class class class)co 7360   supp csupp 8103

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 5231  ax-nul 5241  ax-pr 5370  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 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-supp 8104

This theorem is referenced by:  fsuppunbi  9295  gsumzaddlem  19887