Theorem rnmptbd2 45984

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

Hypotheses

Ref	Expression
rnmptbd2.x	⊢ Ⅎ𝑥𝜑
rnmptbd2.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
rnmptbd2	⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))

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 rnmptbd2

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

Step	Hyp	Ref	Expression
1		breq1 5112	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
2	1	ralbidv 3188	. . . 4 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
3	2	cbvrexvw 3244	. . 3 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
4	3	a1i 11	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
5		rnmptbd2.x	. . 3 ⊢ Ⅎ𝑥𝜑
6		rnmptbd2.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
7	5, 6	rnmptbd2lem 45983	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑢))
8		breq1 5112	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑢 ↔ 𝑦 ≤ 𝑢))
9	8	ralbidv 3188	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑢 ↔ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑢))
10		breq2 5113	. . . . . 6 ⊢ (𝑢 = 𝑧 → (𝑦 ≤ 𝑢 ↔ 𝑦 ≤ 𝑧))
11	10	cbvralvw 3243	. . . . 5 ⊢ (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑢 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
12	9, 11	bitrdi 290	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑢 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
13	12	cbvrexvw 3244	. . 3 ⊢ (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)
14	13	a1i 11	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑢 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))
15	4, 7, 14	3bitrd 308	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  Ⅎwnf 1813   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   class class class wbr 5109   ↦ cmpt 5192  ran crn 5662  ℝcr 11094   ≤ cle 11239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  limsupvaluz2  46472