Theorem rnmptlb 44542

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 2727	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5952	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3475	. . . . 5 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3276	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵
5		nfv 1910	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑧
6		rspa 3240	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑤 ≤ 𝐵)
7	6	3adant3 1130	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝐵)
8		simp3 1136	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
9	7, 8	breqtrrd 5170	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
10	9	3exp 1117	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑤 ≤ 𝑧)))
11	4, 5, 10	rexlimd 3258	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑤 ≤ 𝑧))
12	11	imp 406	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
13	12	adantll 713	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑤 ≤ 𝑧)
14	3, 13	sylan2b 593	. . . 4 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ≤ 𝑧)
15	14	ralrimiva 3141	. . 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 5145	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
18	17	ralbidv 3172	. . . . 5 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
19	18	cbvrexvw 3230	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
20	16, 19	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
21	15, 20	reximddv3 44440	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
22		breq1 5145	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
23	22	ralbidv 3172	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
24	23	cbvrexvw 3230	. 2 ⊢ (∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
25	21, 24	sylib 217	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065  Vcvv 3469   class class class wbr 5142   ↦ cmpt 5225  ran crn 5673  ℝcr 11129   ≤ cle 11271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-mpt 5226  df-cnv 5680  df-dm 5682  df-rn 5683

This theorem is referenced by:  infnsuprnmpt  44549  infrpgernmpt  44770