Theorem rnmptbd 42784

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

Hypotheses

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

Assertion

Ref	Expression
rnmptbd	⊢ (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 rnmptbd

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5083	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑤))
2	1	ralbidv 3123	. . . 4 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤))
3	2	cbvrexvw 3382	. . 3 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤)
4	3	a1i 11	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤))
5		rnmptbd.x	. . 3 ⊢ Ⅎ𝑥𝜑
6		nfv 1921	. . 3 ⊢ Ⅎ𝑤𝜑
7		rnmptbd.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
8	5, 6, 7	rnmptbdlem 42783	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤))
9		breq2 5083	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑢 ≤ 𝑤 ↔ 𝑢 ≤ 𝑦))
10	9	ralbidv 3123	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑦))
11		breq1 5082	. . . . . 6 ⊢ (𝑢 = 𝑧 → (𝑢 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
12	11	cbvralvw 3381	. . . . 5 ⊢ (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑦 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
13	10, 12	bitrdi 287	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
14	13	cbvrexvw 3382	. . 3 ⊢ (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
15	14	a1i 11	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
16	4, 8, 15	3bitrd 305	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1790   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067   class class class wbr 5079   ↦ cmpt 5162  ran crn 5591  ℝcr 10881   ≤ cle 11021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-mpt 5163  df-cnv 5598  df-dm 5600  df-rn 5601

This theorem is referenced by:  supxrre3rnmpt  42951