Theorem rnmptbd2lem 45434

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 2734	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	elrnmpt 5905	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
3	2	elv 3443	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
4		nfra1 3258	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵
5		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
6		rspa 3223	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐵 → 𝐵 = 𝑧)
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ≤ 𝐵 ∧ 𝑧 = 𝐵) → 𝐵 = 𝑧)
11	7, 10	breqtrd 5122	. . . . . . . . . . . 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 3241	. . . . . . . 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 714	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → 𝑦 ≤ 𝑧)
18	3, 17	sylan2b 594	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ≤ 𝑧)
19	18	ralrimiva 3126	. . . 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 3149	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
22		rnmptbd2lem.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
23		nfmpt1 5195	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
24	23	nfrn 5899	. . . . . . 7 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
25	24, 5	nfralw 3281	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧
26	22, 25	nfan 1900	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
27		breq2 5100	. . . . . 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 715	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
32	1, 29, 31	elrnmpt1d 5911	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
33	27, 28, 32	rspcdva 3575	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ≤ 𝐵)
34	26, 33	ralrimia 3233	. . . 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 3149	. 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 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   class class class wbr 5096   ↦ cmpt 5177  ran crn 5623  ℝcr 11023   ≤ cle 11165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-mpt 5178  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  rnmptbd2  45435