Theorem suppss3 30212

Description: Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
suppss3.1	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵)
suppss3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
suppss3.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
suppss3.2	⊢ (𝜑 → 𝐹 Fn 𝐴)
suppss3.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)

Assertion

Ref	Expression
suppss3	⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑍   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem suppss3

Step	Hyp	Ref	Expression
1		suppss3.1	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	oveq1i 6986	. 2 ⊢ (𝐺 supp 𝑍) = ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍)
3		simpl 475	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4		eldifi 3994	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
5	4	adantl 474	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝑥 ∈ 𝐴)
6		suppss3.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		suppss3.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fnex 6806	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
9	6, 7, 8	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
10		suppss3.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		suppimacnv 7644	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	9, 10, 11	syl2anc 576	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	12	eleq2d 2852	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
14		elpreima 6653	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
15	6, 14	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
16	13, 15	bitrd 271	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
17	16	baibd 532	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
18	17	notbid 310	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
19	18	biimpd 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
20	19	expimpd 446	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
21		eldif 3840	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22		fvex 6512	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
23		eldifsn 4593	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ (𝐹‘𝑥) ≠ 𝑍))
24	22, 23	mpbiran 696	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑥) ≠ 𝑍)
25	24	necon2bbii 3019	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑍 ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))
26	20, 21, 25	3imtr4g 288	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹‘𝑥) = 𝑍))
27	26	imp 398	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
28		suppss3.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)
29	3, 5, 27, 28	syl3anc 1351	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
30	29, 7	suppss2 7667	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
31	2, 30	syl5eqss 3906	1 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2968  Vcvv 3416   ∖ cdif 3827   ⊆ wss 3830  {csn 4441   ↦ cmpt 5008  ^◡ccnv 5406   “ cima 5410   Fn wfn 6183  ‘cfv 6188  (class class class)co 6976   supp csupp 7633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-supp 7634

This theorem is referenced by:  eulerpartlems  31260