Theorem rnmptlb 45991

Description: Boundness below of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
rnmptlb.1	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)

Assertion

Ref	Expression
rnmptlb	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝜑,𝑧   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rnmptlb

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5950	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3462	. . . . 5 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3291	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵
5		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑧
6		rspa 3256	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑤 ≤ 𝐵)
7	6	3adant3 1150	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝐵)
8		simp3 1156	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
9	7, 8	breqtrrd 5141	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
10	9	3exp 1137	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ≤ 𝑧)))
11	4, 5, 10	rexlimd 3274	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ≤ 𝑧))
12	11	imp 412	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
13	12	adantll 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
14	3, 13	sylan2b 606	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ≤ 𝑧)
15	14	ralrimiva 3159	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
16		rnmptlb.1	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
17		breq1 5114	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
18	17	ralbidv 3190	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
19	18	cbvrexvw 3246	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
20	16, 19	sylib 221	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
21	15, 20	reximddv3 3184	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
22		breq1 5114	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
23	22	ralbidv 3190	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
24	23	cbvrexvw 3246	. 2 ⊢ (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
25	21, 24	sylib 221	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   class class class wbr 5111   ↦ cmpt 5194  ran crn 5664  ℝcr 11110   ≤ cle 11255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  infnsuprnmpt  45998  infrpgernmpt  46212