Theorem stoweidlem4 44593

Description: Lemma for stoweid 44652: 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 2822	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ))
2	1	anbi2d 630	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝐵 ∈ ℝ)))
3		simpl 484	. . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝐵)
4	3	mpteq2dva 5244	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐵))
5	4	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴))
6	2, 5	imbi12d 345	. . 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 3555	. 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 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5227  ℝcr 11096

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-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-opab 5207  df-mpt 5228

This theorem is referenced by:  stoweidlem18  44607  stoweidlem19  44608  stoweidlem22  44611  stoweidlem32  44621  stoweidlem36  44625  stoweidlem40  44629  stoweidlem41  44630  stoweidlem55  44644