Theorem rnmptbddlem 45208

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

Hypotheses

Ref	Expression
rnmptbddlem.x	⊢ Ⅎ𝑥𝜑
rnmptbddlem.b	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)

Assertion

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

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

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

Proof of Theorem rnmptbddlem

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5949	. . . . 5 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3468	. . . 4 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		rnmptbddlem.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ℝ
6	4, 5	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ)
7		nfra1 3269	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
8	6, 7	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
9		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
10		simp3 1138	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
11		rspa 3234	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
12	11	3adant3 1132	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦)
13	10, 12	eqbrtrd 5145	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
14	13	3exp 1119	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
15	14	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
16	8, 9, 15	rexlimd 3252	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦))
17	16	imp 406	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
18	3, 17	sylan2b 594	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑧 ≤ 𝑦)
19	18	ralrimiva 3133	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
20		rnmptbddlem.b	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
21	19, 20	reximddv3 3159	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  Ⅎwnf 1782   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  Vcvv 3463   class class class wbr 5123   ↦ cmpt 5205  ran crn 5666  ℝcr 11136   ≤ cle 11278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-mpt 5206  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  rnmptbdd  45209