Theorem rnmptbd2lem 46023

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

Hypotheses

Ref	Expression
rnmptbd2lem.x	⊢ Ⅎ𝑥𝜑
rnmptbd2lem.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
rnmptbd2lem	⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))

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

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

Proof of Theorem rnmptbd2lem

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5950	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3462	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3291	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵
5		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
6		rspa 3256	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 𝐵)
7		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 ≤ 𝐵)
8		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
9	8	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
10	9	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝐵 = 𝑧)
11	7, 10	breqtrd 5139	. . . . . . . . . . . 12 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
12	11	ex 418	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝐵 → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
13	6, 12	syl 18	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
14	13	ex 418	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧)))
15	4, 5, 14	rexlimd 3274	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
16	15	imp 412	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
17	16	adantll 727	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
18	3, 17	sylan2b 606	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≤ 𝑧)
19	18	ralrimiva 3159	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
20	19	ex 418	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
21	20	reximdv 3182	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
22		rnmptbd2lem.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
23		nfmpt1 5212	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
24	23	nfrn 5944	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24, 5	nfralw 3314	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧
26	22, 25	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
27		breq2 5115	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝐵))
28		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
29		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
30		rnmptbd2lem.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
31	30	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
32	1, 29, 31	elrnmpt1d 5956	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
33	27, 28, 32	rspcdva 3584	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 𝐵)
34	26, 33	ralrimia 3266	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
35	34	ex 418	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧 → ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵))
36	35	reximdv 3182	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵))
37	21, 36	impbid 215	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   class class class wbr 5111   ↦ cmpt 5194  ran crn 5664  ℝcr 11116   ≤ cle 11261

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  rnmptbd2  46024