Theorem rnmptbdd 45352

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 5093	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑣))
4	3	ralbidv 3155	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣))
5	4	cbvrexvw 3211	. . . 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 45351	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣)
8		breq2 5093	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑤 ≤ 𝑣 ↔ 𝑤 ≤ 𝑦))
9	8	ralbidv 3155	. . . 4 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦))
10		breq1 5092	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
11	10	cbvralvw 3210	. . . 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 3211	. 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 1784  ∀wral 3047  ∃wrex 3056   class class class wbr 5089   ↦ cmpt 5170  ran crn 5615  ℝcr 11005   ≤ cle 11147

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  suprclrnmpt  45358  suprubrnmpt2  45359  suprubrnmpt  45360  rnmptbdlem  45362  supxrrernmpt  45529  suprleubrnmpt  45530  supminfrnmpt  45553