Theorem rnmptbddlem 45602

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 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5915	. . . . 5 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3447	. . . 4 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		rnmptbddlem.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ℝ
6	4, 5	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ)
7		nfra1 3262	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
8	6, 7	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
9		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
10		simp3 1139	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
11		rspa 3227	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
12	11	3adant3 1133	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ≤ 𝑦)
13	10, 12	eqbrtrd 5122	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ≤ 𝑦)
14	13	3exp 1120	. . . . . . 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 3245	. . . . 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 595	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑧 ≤ 𝑦)
19	18	ralrimiva 3130	. 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 3155	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   class class class wbr 5100   ↦ cmpt 5181  ran crn 5633  ℝcr 11037   ≤ cle 11179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-cnv 5640  df-dm 5642  df-rn 5643

This theorem is referenced by:  rnmptbdd  45603