Theorem rnmptlb 45694

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 2740	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5907	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3437	. . . . 5 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3264	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵
5		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑧
6		rspa 3229	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑤 ≤ 𝐵)
7	6	3adant3 1138	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝐵)
8		simp3 1144	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
9	7, 8	breqtrrd 5107	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
10	9	3exp 1125	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ≤ 𝑧)))
11	4, 5, 10	rexlimd 3247	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ≤ 𝑧))
12	11	imp 407	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
13	12	adantll 720	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
14	3, 13	sylan2b 600	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ≤ 𝑧)
15	14	ralrimiva 3132	. . 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 5082	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
18	17	ralbidv 3163	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
19	18	cbvrexvw 3219	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
20	16, 19	sylib 219	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
21	15, 20	reximddv3 3157	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
22		breq1 5082	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
23	22	ralbidv 3163	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
24	23	cbvrexvw 3219	. 2 ⊢ (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
25	21, 24	sylib 219	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   class class class wbr 5079   ↦ cmpt 5160  ran crn 5626  ℝcr 11035   ≤ cle 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-mpt 5161  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  infnsuprnmpt  45701  infrpgernmpt  45915