Theorem rnmptbddlem 44033

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 2732	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5955	. . . . 5 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3480	. . . 4 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		rnmptbddlem.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 ∈ ℝ
6	4, 5	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ℝ)
7		nfra1 3281	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
8	6, 7	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
9		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
10		simp3 1138	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
11		rspa 3245	. . . . . . . . . 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 5170	. . . . . . . 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 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ≤ 𝑦)))
16	8, 9, 15	rexlimd 3263	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ≤ 𝑦))
17	16	imp 407	. . . 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 3146	. 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 43928	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  ℝcr 11111   ≤ cle 11251

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by:  rnmptbdd  44034