Theorem psrbagfsuppub 15192

Description: Given a finitely supported function on finite bags, there is an upper bound of values which can map to anything other than zero. (Contributed by Jim Kingdon, 5-Nov-2026.)

Hypotheses

Ref	Expression
psrbagfsuppub.f	⊢ (𝜑 → 𝐹:(ℕ0 ↑𝑚 𝐼)⟶𝐵)
psrbagfsuppub.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
psrbagfsuppub.i	⊢ (𝜑 → 𝐼 ∈ Fin)
psrbagfsuppub.s	⊢ (𝜑 → 𝐹 finSupp 𝑍)

Assertion

Ref	Expression
psrbagfsuppub	⊢ (𝜑 → ∃𝑘 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑘 → (𝐹‘𝑦) = 𝑍))

Distinct variable groups:   𝑦,𝐹,𝑘   𝑦,𝐼,𝑘   𝑦,𝑍,𝑘   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑘)   𝐵(𝑦, 𝑘)

Proof of Theorem psrbagfsuppub

Dummy variables 𝑗 𝑝 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrbagfsuppub.s	. . 3 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
2	1	fsuppfiss 7301	. 2 ⊢ (𝜑 → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))
3		simpr2 1035	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ⊆ dom 𝐹)
4		psrbagfsuppub.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(ℕ0 ↑𝑚 𝐼)⟶𝐵)
5	4	fdmd 5543	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = (ℕ0 ↑𝑚 𝐼))
6	5	sseq2d 3278	. . . . . 6 ⊢ (𝜑 → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ (ℕ0 ↑𝑚 𝐼)))
7	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ (ℕ0 ↑𝑚 𝐼)))
8	3, 7	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ⊆ (ℕ0 ↑𝑚 𝐼))
9		simpr1 1034	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ∈ Fin)
10		psrbagfsuppub.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin)
11	10	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝐼 ∈ Fin)
12	8, 9, 11	psrbagmaxlem1 15185	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → ∃𝑝(𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗)))
13		simprl 535	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → 𝑝:𝐼⟶ℕ0)
14		nn0ex 9581	. . . . . . 7 ⊢ ℕ0 ∈ V
15	14	a1i 9	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ℕ0 ∈ V)
16	11	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → 𝐼 ∈ Fin)
17	15, 16	elmapd 6941	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → (𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ↔ 𝑝:𝐼⟶ℕ0))
18	13, 17	mpbird 167	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → 𝑝 ∈ (ℕ0 ↑𝑚 𝐼))
19		simpr3 1036	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})
20	19	ad3antrrr 496	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})
21	4	ffnd 5537	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn (ℕ0 ↑𝑚 𝐼))
22	21	ad4antr 498	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → 𝐹 Fn (ℕ0 ↑𝑚 𝐼))
23		simplr 533	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → 𝑦 ∈ (ℕ0 ↑𝑚 𝐼))
24	5	ad4antr 498	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → dom 𝐹 = (ℕ0 ↑𝑚 𝐼))
25	23, 24	eleqtrrd 2318	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → 𝑦 ∈ dom 𝐹)
26		fveq1 5697	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥‘𝑗) = (𝑦‘𝑗))
27	26	breq1d 4141	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ (𝑦‘𝑗) ≤ (𝑝‘𝑗)))
28	27	ralbidv 2550	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (∀𝑗 ∈ 𝐼 (𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑗 ∈ 𝐼 (𝑦‘𝑗) ≤ (𝑝‘𝑗)))
29		simprr 537	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))
30	29	ad3antrrr 496	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))
31		ralcom 2714	. . . . . . . . . . . . . 14 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑥 ∈ 𝑤 ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) ≤ (𝑝‘𝑗))
32	30, 31	sylib 122	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → ∀𝑥 ∈ 𝑤 ∀𝑗 ∈ 𝐼 (𝑥‘𝑗) ≤ (𝑝‘𝑗))
33		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝑦 ∈ 𝑤)
34	28, 32, 33	rspcdva 2934	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → ∀𝑗 ∈ 𝐼 (𝑦‘𝑗) ≤ (𝑝‘𝑗))
35		simp-5l 549	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝜑)
36	23	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝑦 ∈ (ℕ0 ↑𝑚 𝐼))
37		elmapi 6949	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑦:𝐼⟶ℕ0)
38	36, 37	syl 14	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝑦:𝐼⟶ℕ0)
39	13	ad3antrrr 496	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝑝:𝐼⟶ℕ0)
40		simp2 1029	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → 𝑦:𝐼⟶ℕ0)
41	40	ffnd 5537	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → 𝑦 Fn 𝐼)
42		simp3 1030	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → 𝑝:𝐼⟶ℕ0)
43	42	ffnd 5537	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → 𝑝 Fn 𝐼)
44	10	3ad2ant1 1049	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → 𝐼 ∈ Fin)
45		inidm 3440	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∩ 𝐼) = 𝐼
46		eqidd 2239	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) ∧ 𝑗 ∈ 𝐼) → (𝑦‘𝑗) = (𝑦‘𝑗))
47		eqidd 2239	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) ∧ 𝑗 ∈ 𝐼) → (𝑝‘𝑗) = (𝑝‘𝑗))
48	41, 43, 44, 44, 45, 46, 47	ofrfval 6315	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦:𝐼⟶ℕ0 ∧ 𝑝:𝐼⟶ℕ0) → (𝑦 ∘𝑟 ≤ 𝑝 ↔ ∀𝑗 ∈ 𝐼 (𝑦‘𝑗) ≤ (𝑝‘𝑗)))
49	35, 38, 39, 48	syl3anc 1278	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → (𝑦 ∘𝑟 ≤ 𝑝 ↔ ∀𝑗 ∈ 𝐼 (𝑦‘𝑗) ≤ (𝑝‘𝑗)))
50	34, 49	mpbird 167	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → 𝑦 ∘𝑟 ≤ 𝑝)
51		simplr 533	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) ∧ 𝑦 ∈ 𝑤) → ¬ 𝑦 ∘𝑟 ≤ 𝑝)
52	50, 51	pm2.65da 671	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → ¬ 𝑦 ∈ 𝑤)
53	25, 52	eldifd 3230	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → 𝑦 ∈ (dom 𝐹 ∖ 𝑤))
54	22, 23, 53	fnfvimad 5957	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → (𝐹‘𝑦) ∈ (𝐹 “ (dom 𝐹 ∖ 𝑤)))
55	20, 54	sseldd 3249	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → (𝐹‘𝑦) ∈ {𝑍})
56		elsni 3727	. . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {𝑍} → (𝐹‘𝑦) = 𝑍)
57	55, 56	syl 14	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ ¬ 𝑦 ∘𝑟 ≤ 𝑝) → (𝐹‘𝑦) = 𝑍)
58	57	ex 115	. . . . 5 ⊢ ((((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ 𝑦 ∈ (ℕ0 ↑𝑚 𝐼)) → (¬ 𝑦 ∘𝑟 ≤ 𝑝 → (𝐹‘𝑦) = 𝑍))
59	58	ralrimiva 2623	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑝 → (𝐹‘𝑦) = 𝑍))
60		breq2 4135	. . . . . 6 ⊢ (𝑘 = 𝑝 → (𝑦 ∘𝑟 ≤ 𝑘 ↔ 𝑦 ∘𝑟 ≤ 𝑝))
61	60	notbid 677	. . . . 5 ⊢ (𝑘 = 𝑝 → (¬ 𝑦 ∘𝑟 ≤ 𝑘 ↔ ¬ 𝑦 ∘𝑟 ≤ 𝑝))
62	61	rspceaimv 2938	. . . 4 ⊢ ((𝑝 ∈ (ℕ0 ↑𝑚 𝐼) ∧ ∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑝 → (𝐹‘𝑦) = 𝑍)) → ∃𝑘 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑘 → (𝐹‘𝑦) = 𝑍))
63	18, 59, 62	syl2anc 415	. . 3 ⊢ (((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑤 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ∃𝑘 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑘 → (𝐹‘𝑦) = 𝑍))
64	12, 63	exlimddv 1954	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → ∃𝑘 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑘 → (𝐹‘𝑦) = 𝑍))
65	2, 64	exlimddv 1954	1 ⊢ (𝜑 → ∃𝑘 ∈ (ℕ0 ↑𝑚 𝐼)∀𝑦 ∈ (ℕ0 ↑𝑚 𝐼)(¬ 𝑦 ∘𝑟 ≤ 𝑘 → (𝐹‘𝑦) = 𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∖ cdif 3217   ⊆ wss 3220  {csn 3709   class class class wbr 4131  dom cdm 4776   “ cima 4779   Fn wfn 5375  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   ∘_𝑟 cofr 6304   ↑_𝑚 cmap 6927  Fincfn 7028   finSupp cfsupp 7292   ≤ cle 8368  ℕ₀cn0 9575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-ofr 6306  df-1st 6378  df-2nd 6379  df-recs 6581  df-frec 6667  df-er 6812  df-map 6929  df-en 7029  df-fin 7031  df-fsuppfi 7293  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9655  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787

This theorem is used by:  mplsubrglemcl  15249