Theorem suppun 8186

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 4127	. . . . . 6 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
2		cnvun 6137	. . . . . . . 8 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
3	2	imaeq1i 6057	. . . . . . 7 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
4		imaundir 6146	. . . . . . 7 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
5	3, 4	eqtri 2785	. . . . . 6 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
6	1, 5	sseqtrri 3983	. . . . 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 8176	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
9	8	adantr 486	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
10		suppun.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
11		unexg 7749	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
12	11	adantlr 728	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
13	10, 12	sylan2 605	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 ∪ 𝐺) ∈ V)
14		simplr 781	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15		suppimacnv 8176	. . . . 5 ⊢ (((𝐹 ∪ 𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
16	13, 14, 15	syl2anc 596	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹 ∪ 𝐺) supp 𝑍) = (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})))
17	7, 9, 16	3sstr4d 3989	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
18	17	ex 418	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)))
19		supp0prc 8165	. . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20		0ss 4353	. . . 4 ⊢ ∅ ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)
21	19, 20	eqsstrdi 3978	. . 3 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
22	21	a1d 26	. 2 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)))
23	18, 22	pm2.61i 184	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587  ^◡ccnv 5658   “ cima 5662  (class class class)co 7417   supp csupp 8162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-supp 8163

This theorem is used by:  fsuppunbi  9363  gsumzaddlem  20054