Theorem rnmptlb 45958

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 2763	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5948	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3460	. . . . 5 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3289	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵
5		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑧
6		rspa 3254	. . . . . . . . . . 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 5139	. . . . . . . . 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 3272	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ≤ 𝑧))
12	11	imp 411	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
13	12	adantll 726	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
14	3, 13	sylan2b 605	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ≤ 𝑧)
15	14	ralrimiva 3157	. . 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 5112	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
18	17	ralbidv 3188	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
19	18	cbvrexvw 3244	. . . 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 3182	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
22		breq1 5112	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
23	22	ralbidv 3188	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
24	23	cbvrexvw 3244	. 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   class class class wbr 5109   ↦ cmpt 5192  ran crn 5662  ℝcr 11094   ≤ cle 11239

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-sep 5257  ax-pr 5404

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-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  infnsuprnmpt  45965  infrpgernmpt  46179