Theorem rnmptbdd 41522

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

Hypotheses

Ref	Expression
rnmptbdd.x	⊢ Ⅎ𝑥𝜑
rnmptbdd.b	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)

Assertion

Ref	Expression
rnmptbdd	⊢ (𝜑 → ∃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)

Proof of Theorem rnmptbdd

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

Step	Hyp	Ref	Expression
1		rnmptbdd.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		rnmptbdd.b	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
3		breq2 5073	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑣))
4	3	ralbidv 3200	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣))
5	4	cbvrexvw 3453	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
6	2, 5	sylib 220	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
7	1, 6	rnmptbddlem 41521	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣)
8		breq2 5073	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑤 ≤ 𝑣 ↔ 𝑤 ≤ 𝑦))
9	8	ralbidv 3200	. . . 4 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦))
10		breq1 5072	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
11	10	cbvralvw 3452	. . . 4 ⊢ (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
12	9, 11	syl6bb 289	. . 3 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
13	12	cbvrexvw 3453	. 2 ⊢ (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
14	7, 13	sylib 220	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4  Ⅎwnf 1783  ∀wral 3141  ∃wrex 3142   class class class wbr 5069   ↦ cmpt 5149  ran crn 5559  ℝcr 10539   ≤ cle 10679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-mpt 5150  df-cnv 5566  df-dm 5568  df-rn 5569

This theorem is referenced by:  suprclrnmpt  41529  suprubrnmpt2  41530  suprubrnmpt  41531  rnmptbdlem  41533  supxrrernmpt  41701  suprleubrnmpt  41702