Theorem suppun 8169

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 4173	. . . . . 6 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
2		cnvun 6143	. . . . . . . 8 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
3	2	imaeq1i 6057	. . . . . . 7 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍}))
4		imaundir 6151	. . . . . . 7 ⊢ ((◡𝐹 ∪ ◡𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
5	3, 4	eqtri 2761	. . . . . 6 ⊢ (◡(𝐹 ∪ 𝐺) “ (V ∖ {𝑍})) = ((◡𝐹 “ (V ∖ {𝑍})) ∪ (◡𝐺 “ (V ∖ {𝑍})))
6	1, 5	sseqtrri 4020	. . . . 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 8159	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
9	8	adantr 482	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
10		suppun.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
11		unexg 7736	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ 𝑉) → (𝐹 ∪ 𝐺) ∈ V)
12	11	adantlr 714	. . . . . 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 768	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15		suppimacnv 8159	. . . . 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 4030	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
18	17	ex 414	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)))
19		supp0prc 8149	. . . 4 ⊢ (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20		0ss 4397	. . . 4 ⊢ ∅ ⊆ ((𝐹 ∪ 𝐺) supp 𝑍)
21	19, 20	eqsstrdi 4037	. . 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947   ⊆ wss 3949  ∅c0 4323  {csn 4629  ^◡ccnv 5676   “ cima 5680  (class class class)co 7409   supp csupp 8146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-supp 8147

This theorem is referenced by:  fsuppunbi  9384  gsumzaddlem  19789