Theorem stoweidlem4 45925

Description: Lemma for stoweid 45984: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypothesis

Ref	Expression
stoweidlem4.1	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem4	⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)

Distinct variable groups:   𝑥,𝑡,𝐵   𝑥,𝐴   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑇(𝑡)

Proof of Theorem stoweidlem4

Step	Hyp	Ref	Expression
1		eleq1 2832	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
2	1	anbi2d 629	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝐵 ∈ ℝ)))
3		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝐵)
4	3	mpteq2dva 5266	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐵))
5	4	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴))
6	2, 5	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐵 → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)))
7		stoweidlem4.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
8	6, 7	vtoclg 3566	. 2 ⊢ (𝐵 ∈ ℝ → ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴))
9	8	anabsi7 670	1 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  ℝcr 11183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-opab 5229  df-mpt 5250

This theorem is referenced by:  stoweidlem18  45939  stoweidlem19  45940  stoweidlem22  45943  stoweidlem32  45953  stoweidlem36  45957  stoweidlem40  45961  stoweidlem41  45962  stoweidlem55  45976