Theorem rnmptbd2lem 45157

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 2740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5981	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3493	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3290	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵
5		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
6		rspa 3254	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 𝐵)
7		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 ≤ 𝐵)
8		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
9	8	eqcomd 2746	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝐵 = 𝑧)
11	7, 10	breqtrd 5192	. . . . . . . . . . . 12 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
12	11	ex 412	. . . . . . . . . . 11 ⊢ (𝑦 ≤ 𝐵 → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
13	6, 12	syl 17	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
14	13	ex 412	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑦 ≤ 𝑧)))
15	4, 5, 14	rexlimd 3272	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑦 ≤ 𝑧))
16	15	imp 406	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
17	16	adantll 713	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
18	3, 17	sylan2b 593	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≤ 𝑧)
19	18	ralrimiva 3152	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
20	19	ex 412	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
21	20	reximdv 3176	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
22		rnmptbd2lem.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
23		nfmpt1 5274	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
24	23	nfrn 5977	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24, 5	nfralw 3317	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧
26	22, 25	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
27		breq2 5170	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝐵))
28		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
29		simpr 484	. . . . . . 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 714	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
32	1, 29, 31	elrnmpt1d 5987	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
33	27, 28, 32	rspcdva 3636	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 𝐵)
34	26, 33	ralrimia 3264	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
35	34	ex 412	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧 → ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵))
36	35	reximdv 3176	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵))
37	21, 36	impbid 212	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701  ℝcr 11183   ≤ cle 11325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  rnmptbd2  45158