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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem4 | Structured version Visualization version GIF version |
Description: Lemma for stoweid 44769: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem4.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
Ref | Expression |
---|---|
stoweidlem4 | ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2821 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ)) | |
2 | 1 | anbi2d 629 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝐵 ∈ ℝ))) |
3 | simpl 483 | . . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝐵) | |
4 | 3 | mpteq2dva 5248 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐵)) |
5 | 4 | eleq1d 2818 | . . . 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 3556 | . 2 ⊢ (𝐵 ∈ ℝ → ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)) |
9 | 8 | anabsi7 669 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5231 ℝcr 11108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-opab 5211 df-mpt 5232 |
This theorem is referenced by: stoweidlem18 44724 stoweidlem19 44725 stoweidlem22 44728 stoweidlem32 44738 stoweidlem36 44742 stoweidlem40 44746 stoweidlem41 44747 stoweidlem55 44761 |
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