Theorem psrbagmax 15183

Description: Given two finite sets of bags, provide a finite set which contains all of their elements.

The proof constructs the result (𝑍 in the lemmas), proves its properties (via psrbagmaxlemfi 15180, psrbagmaxlem1x 15181, and psrbagmaxlem1y 15182), and then this proof combines those lemmas and assigns the result to the variable 𝑧. (Contributed by Jim Kingdon, 24-Oct-2026.)

Hypotheses

Ref	Expression
psrbagmax.x	⊢ (𝜑 → 𝑋 ⊆ (ℕ0 ↑𝑚 𝐼))
psrbagmax.xfi	⊢ (𝜑 → 𝑋 ∈ Fin)
psrbagmax.y	⊢ (𝜑 → 𝑌 ⊆ (ℕ0 ↑𝑚 𝐼))
psrbagmax.yfi	⊢ (𝜑 → 𝑌 ∈ Fin)
psrbagmax.i	⊢ (𝜑 → 𝐼 ∈ Fin)

Assertion

Ref	Expression
psrbagmax	⊢ (𝜑 → ∃𝑧((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧)))

Distinct variable groups:   𝑧,𝐼   𝑧,𝑋   𝑧,𝑌

Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem psrbagmax

Dummy variables 𝑖 𝑗 𝑘 𝑝 𝑟 𝑢 𝑥 𝑞 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrbagmax.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (ℕ0 ↑𝑚 𝐼))
2		psrbagmax.xfi	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
3		psrbagmax.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
4	1, 2, 3	psrbagmaxlem1 15178	. 2 ⊢ (𝜑 → ∃𝑝(𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗)))
5		psrbagmax.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ (ℕ0 ↑𝑚 𝐼))
6		psrbagmax.yfi	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Fin)
7	5, 6, 3	psrbagmaxlem1 15178	. . . 4 ⊢ (𝜑 → ∃𝑟(𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗)))
8	7	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ∃𝑟(𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗)))
9	1	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑋 ⊆ (ℕ0 ↑𝑚 𝐼))
10	2	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑋 ∈ Fin)
11	3	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝐼 ∈ Fin)
12		simplrl 541	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑝:𝐼⟶ℕ0)
13		simprl 535	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑟:𝐼⟶ℕ0)
14		eqid 2238	. . . . 5 ⊢ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < )) = (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))
15		eqid 2238	. . . . 5 ⊢ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))}
16	9, 10, 11, 12, 13, 14, 15	psrbagmaxlemfi 15180	. . . 4 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin)
17		ssrab2 3333	. . . . . 6 ⊢ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼)
18	16, 17	jctil 312	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼) ∧ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin))
19		simplrr 542	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))
20		fveq1 5697	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑥‘𝑗) = (𝑢‘𝑗))
21	20	breq1d 4141	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ (𝑢‘𝑗) ≤ (𝑝‘𝑗)))
22	21	cbvralv 2786	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑢 ∈ 𝑋 (𝑢‘𝑗) ≤ (𝑝‘𝑗))
23	22	ralbii 2556	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑗 ∈ 𝐼 ∀𝑢 ∈ 𝑋 (𝑢‘𝑗) ≤ (𝑝‘𝑗))
24		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑢‘𝑗) = (𝑢‘𝑘))
25		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑝‘𝑗) = (𝑝‘𝑘))
26	24, 25	breq12d 4144	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → ((𝑢‘𝑗) ≤ (𝑝‘𝑗) ↔ (𝑢‘𝑘) ≤ (𝑝‘𝑘)))
27	26	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (∀𝑢 ∈ 𝑋 (𝑢‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑢 ∈ 𝑋 (𝑢‘𝑘) ≤ (𝑝‘𝑘)))
28	27	cbvralv 2786	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑢 ∈ 𝑋 (𝑢‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ 𝑋 (𝑢‘𝑘) ≤ (𝑝‘𝑘))
29	23, 28	bitri 184	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗) ↔ ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ 𝑋 (𝑢‘𝑘) ≤ (𝑝‘𝑘))
30	19, 29	sylib 122	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ 𝑋 (𝑢‘𝑘) ≤ (𝑝‘𝑘))
31		simprr 537	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))
32		fveq1 5697	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝑦‘𝑗) = (𝑣‘𝑗))
33	32	breq1d 4141	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝑦‘𝑗) ≤ (𝑟‘𝑗) ↔ (𝑣‘𝑗) ≤ (𝑟‘𝑗)))
34	33	cbvralv 2786	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗) ↔ ∀𝑣 ∈ 𝑌 (𝑣‘𝑗) ≤ (𝑟‘𝑗))
35	34	ralbii 2556	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗) ↔ ∀𝑗 ∈ 𝐼 ∀𝑣 ∈ 𝑌 (𝑣‘𝑗) ≤ (𝑟‘𝑗))
36		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑣‘𝑗) = (𝑣‘𝑘))
37		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑟‘𝑗) = (𝑟‘𝑘))
38	36, 37	breq12d 4144	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → ((𝑣‘𝑗) ≤ (𝑟‘𝑗) ↔ (𝑣‘𝑘) ≤ (𝑟‘𝑘)))
39	38	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (∀𝑣 ∈ 𝑌 (𝑣‘𝑗) ≤ (𝑟‘𝑗) ↔ ∀𝑣 ∈ 𝑌 (𝑣‘𝑘) ≤ (𝑟‘𝑘)))
40	39	cbvralv 2786	. . . . . . . 8 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑣 ∈ 𝑌 (𝑣‘𝑗) ≤ (𝑟‘𝑗) ↔ ∀𝑘 ∈ 𝐼 ∀𝑣 ∈ 𝑌 (𝑣‘𝑘) ≤ (𝑟‘𝑘))
41	35, 40	bitri 184	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗) ↔ ∀𝑘 ∈ 𝐼 ∀𝑣 ∈ 𝑌 (𝑣‘𝑘) ≤ (𝑟‘𝑘))
42	31, 41	sylib 122	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ∀𝑘 ∈ 𝐼 ∀𝑣 ∈ 𝑌 (𝑣‘𝑘) ≤ (𝑟‘𝑘))
43	5	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑌 ⊆ (ℕ0 ↑𝑚 𝐼))
44	6	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑌 ∈ Fin)
45	9, 10, 11, 12, 13, 14, 15, 30, 42, 43, 44	psrbagmaxlem1x 15181	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))})
46	9, 10, 11, 12, 13, 14, 15, 30, 42, 43, 44	psrbagmaxlem1y 15182	. . . . 5 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))})
47	18, 45, 46	jca32 310	. . . 4 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → (({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼) ∧ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin) ∧ (𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∧ 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))})))
48		sseq1 3271	. . . . . . 7 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → (𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ↔ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼)))
49		eleq1 2301	. . . . . . 7 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → (𝑧 ∈ Fin ↔ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin))
50	48, 49	anbi12d 477	. . . . . 6 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → ((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ↔ ({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼) ∧ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin)))
51		sseq2 3272	. . . . . . 7 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))}))
52		sseq2 3272	. . . . . . 7 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → (𝑌 ⊆ 𝑧 ↔ 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))}))
53	51, 52	anbi12d 477	. . . . . 6 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → ((𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧) ↔ (𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∧ 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))})))
54	50, 53	anbi12d 477	. . . . 5 ⊢ (𝑧 = {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} → (((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧)) ↔ (({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼) ∧ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin) ∧ (𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∧ 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))}))))
55	54	spcegv 2913	. . . 4 ⊢ ({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin → ((({𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ⊆ (ℕ0 ↑𝑚 𝐼) ∧ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∈ Fin) ∧ (𝑋 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))} ∧ 𝑌 ⊆ {𝑞 ∈ (ℕ0 ↑𝑚 𝐼) ∣ 𝑞 ∘𝑟 ≤ (𝑖 ∈ 𝐼 ↦ sup({(𝑝‘𝑖), (𝑟‘𝑖)}, ℝ, < ))})) → ∃𝑧((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧))))
56	16, 47, 55	sylc 62	. . 3 ⊢ (((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) ∧ (𝑟:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑦 ∈ 𝑌 (𝑦‘𝑗) ≤ (𝑟‘𝑗))) → ∃𝑧((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧)))
57	8, 56	exlimddv 1954	. 2 ⊢ ((𝜑 ∧ (𝑝:𝐼⟶ℕ0 ∧ ∀𝑗 ∈ 𝐼 ∀𝑥 ∈ 𝑋 (𝑥‘𝑗) ≤ (𝑝‘𝑗))) → ∃𝑧((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧)))
58	4, 57	exlimddv 1954	1 ⊢ (𝜑 → ∃𝑧((𝑧 ⊆ (ℕ0 ↑𝑚 𝐼) ∧ 𝑧 ∈ Fin) ∧ (𝑋 ⊆ 𝑧 ∧ 𝑌 ⊆ 𝑧)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  {cpr 3710   class class class wbr 4131   ↦ cmpt 4193  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   ∘_𝑟 cofr 6304   ↑_𝑚 cmap 6927  Fincfn 7028  supcsup 7328  ℝcr 8184   < clt 8366   ≤ cle 8367  ℕ₀cn0 9574

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 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

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-1o 6692  df-er 6812  df-map 6929  df-en 7029  df-fin 7031  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9654  df-uz 9937  df-rp 10071  df-fz 10428  df-seqfrec 10905  df-exp 10996  df-cj 11628  df-re 11629  df-im 11630  df-rsqrt 11785  df-abs 11786

This theorem is used by:  mplsubgfilemcl  15232