Theorem suppss3 32802

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 7368	. 2 ⊢ (𝐺 supp 𝑍) = ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍)
3		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝜑)
4		eldifi 4083	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
5	4	adantl 481	. . . 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 7163	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
9	6, 7, 8	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
10		suppss3.z	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
11		suppimacnv 8116	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
12	9, 10, 11	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
13	12	eleq2d 2822	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ 𝑥 ∈ (◡𝐹 “ (V ∖ {𝑍}))))
14		elpreima 7003	. . . . . . . . . . . 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 279	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))))
17	16	baibd 539	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
18	17	notbid 318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
19	18	biimpd 229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ (𝐹 supp 𝑍) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
20	19	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍})))
21		eldif 3911	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐹 supp 𝑍)))
22		fvex 6847	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
23		eldifsn 4742	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑥) ∈ V ∧ (𝐹‘𝑥) ≠ 𝑍))
24	22, 23	mpbiran 709	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑥) ≠ 𝑍)
25	24	necon2bbii 2983	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑍 ↔ ¬ (𝐹‘𝑥) ∈ (V ∖ {𝑍}))
26	20, 21, 25	3imtr4g 296	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → (𝐹‘𝑥) = 𝑍))
27	26	imp 406	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → 𝐵 = 𝑍)
30	29, 7	suppss2 8142	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
31	2, 30	eqsstrid 3972	1 ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  {csn 4580   ↦ cmpt 5179  ^◡ccnv 5623   “ cima 5627   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   supp csupp 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-supp 8103

This theorem is referenced by:  evls1fldgencl  33827  eulerpartlems  34517