Theorem rnmptbdd 45695

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 5090	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑣))
4	3	ralbidv 3161	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣))
5	4	cbvrexvw 3217	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
6	2, 5	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
7	1, 6	rnmptbddlem 45694	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣)
8		breq2 5090	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑤 ≤ 𝑣 ↔ 𝑤 ≤ 𝑦))
9	8	ralbidv 3161	. . . 4 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦))
10		breq1 5089	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
11	10	cbvralvw 3216	. . . 4 ⊢ (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
12	9, 11	bitrdi 287	. . 3 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
13	12	cbvrexvw 3217	. 2 ⊢ (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
14	7, 13	sylib 218	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4  Ⅎwnf 1785  ∀wral 3052  ∃wrex 3062   class class class wbr 5086   ↦ cmpt 5167  ran crn 5626  ℝcr 11031   ≤ cle 11174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  suprclrnmpt  45701  suprubrnmpt2  45702  suprubrnmpt  45703  rnmptbdlem  45705  supxrrernmpt  45870  suprleubrnmpt  45871  supminfrnmpt  45894