Theorem suppss3 33068

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 7420	. 2 ⊢ (𝐺 supp 𝑍) = ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍)
3		simpl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4		eldifi 4085	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
5	4	adantl 486	. . . 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 7215	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
9	6, 7, 8	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
10		suppss3.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		suppimacnv 8166	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	9, 10, 11	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	12	eleq2d 2849	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
14		elpreima 7053	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
15	6, 14	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍})) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
16	13, 15	bitrd 282	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
17	16	baibd 548	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
18	17	notbid 321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
19	18	biimpd 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
20	19	expimpd 458	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
21		eldif 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22		fvex 6894	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
23		eldifsn 4753	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ (𝐹‘𝑥) ≠ 𝑍))
24	22, 23	mpbiran 721	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑥) ≠ 𝑍)
25	24	necon2bbii 3009	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑍 ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))
26	20, 21, 25	3imtr4g 299	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹‘𝑥) = 𝑍))
27	26	imp 411	. . . 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 1398	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
30	29, 7	suppss2 8192	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
31	2, 30	eqsstrid 3975	1 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  {csn 4589   ↦ cmpt 5192  ^◡ccnv 5660   “ cima 5664   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   supp csupp 8152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-supp 8153

This theorem is referenced by:  evls1fldgencl  34060  eulerpartlems  34750