Theorem rnmptbd 39471

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

Hypotheses

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

Assertion

Ref	Expression
rnmptbd	⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃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)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem rnmptbd

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

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑤))
2	1	ralbidv 2986	. . . 4 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤))
3	2	cbvrexv 3172	. . 3 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤)
4	3	a1i 11	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤))
5		rnmptbd.x	. . 3 ⊢ Ⅎ𝑥𝜑
6		nfv 1843	. . 3 ⊢ Ⅎ𝑤𝜑
7		rnmptbd.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
8	5, 6, 7	rnmptbdlem 39470	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤))
9		breq2 4657	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑢 ≤ 𝑤 ↔ 𝑢 ≤ 𝑦))
10	9	ralbidv 2986	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑦))
11		breq1 4656	. . . . . . 7 ⊢ (𝑢 = 𝑧 → (𝑢 ≤ 𝑦 ↔ 𝑧 ≤ 𝑦))
12	11	cbvralv 3171	. . . . . 6 ⊢ (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑦 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
13	12	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑦 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
14	10, 13	bitrd 268	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
15	14	cbvrexv 3172	. . 3 ⊢ (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
16	15	a1i 11	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ℝ ∀𝑢 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑢 ≤ 𝑤 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
17	4, 8, 16	3bitrd 294	1 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  ℝcr 9935   ≤ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  supxrre3rnmpt  39656  supminfrnmpt  39672