Theorem rnmptbdlem 41893

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 2955	. . . . . 6 ⊢ Ⅎ𝑥ℝ
3		nfra1 3183	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
4	2, 3	nfrex 3268	. . . . 5 ⊢ Ⅎ𝑥∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
5	1, 4	nfan 1900	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
6		simpr 488	. . . 4 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
7	5, 6	rnmptbdd 41882	. . 3 ⊢ ((𝜑 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
8	7	ex 416	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
9		rnmptbdlem.y	. . 3 ⊢ Ⅎ𝑦𝜑
10		nfmpt1 5128	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
11	10	nfrn 5788	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
13	11, 12	nfralw 3189	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
14	1, 13	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
15		breq1 5033	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
16		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
17		eqid 2798	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
18		simpr 488	. . . . . . . 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 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
21	17, 18, 20	elrnmpt1d 41866	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
22	15, 16, 21	rspcdva 3573	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
23	14, 22	ralrimia 41767	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
24	23	ex 416	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
25	24	a1d 25	. . 3 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)))
26	9, 25	reximdai 3270	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030   ↦ cmpt 5110  ran crn 5520  ℝcr 10525   ≤ cle 10665

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by:  rnmptbd  41894