Theorem rnmptbdlem 46540

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

Hypotheses

Ref	Expression
rnmptbdlem.x	⊢ Ⅎ𝑥𝜑
rnmptbdlem.y	⊢ Ⅎ𝑦𝜑
rnmptbdlem.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

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

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

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

Proof of Theorem rnmptbdlem

Step	Hyp	Ref	Expression
1		rnmptbdlem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
2		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥ℝ
3		nfra1 3290	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
4	2, 3	nfrexw 3314	. . . . 5 ⊢ Ⅎ𝑥∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
5	1, 4	nfan 1934	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
6		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
7	5, 6	rnmptbdd 46530	. . 3 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
8	7	ex 418	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
9		rnmptbdlem.y	. . 3 ⊢ Ⅎ𝑦𝜑
10		nfmpt1 5210	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
11	10	nfrn 5946	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
13	11, 12	nfralw 3313	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
14	1, 13	nfan 1934	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
15		breq1 5111	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
16		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
17		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
18		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19		rnmptbdlem.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
20	19	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
21	17, 18, 20	elrnmpt1d 5958	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
22	15, 16, 21	rspcdva 3581	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
23	14, 22	ralrimia 3265	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
24	23	ex 418	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
25	24	a1d 26	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)))
26	9, 25	reximdai 3268	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
27	8, 26	impbid 215	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   class class class wbr 5108   ↦ cmpt 5192  ran crn 5659  ℝcr 11238   ≤ cle 11383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-cnv 5666  df-dm 5668  df-rn 5669

This theorem is used by:  rnmptbd  46541