Theorem rnmptbdd 43949

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 5153	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑣))
4	3	ralbidv 3178	. . . . 5 ⊢ (𝑦 = 𝑣 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣))
5	4	cbvrexvw 3236	. . . 4 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
6	2, 5	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑣)
7	1, 6	rnmptbddlem 43948	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣)
8		breq2 5153	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑤 ≤ 𝑣 ↔ 𝑤 ≤ 𝑦))
9	8	ralbidv 3178	. . . 4 ⊢ (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑦))
10		breq1 5152	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
11	10	cbvralvw 3235	. . . 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 3236	. 2 ⊢ (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
14	7, 13	sylib 217	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)

Syntax hints:   → wi 4  Ⅎwnf 1786  ∀wral 3062  ∃wrex 3071   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678  ℝcr 11109   ≤ cle 11249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688

This theorem is referenced by:  suprclrnmpt  43955  suprubrnmpt2  43956  suprubrnmpt  43957  rnmptbdlem  43959  supxrrernmpt  44131  suprleubrnmpt  44132  supminfrnmpt  44155