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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem4 | Structured version Visualization version GIF version |
Description: Lemma for stoweid 42355: 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 2902 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ ℝ ↔ 𝐵 ∈ ℝ)) | |
2 | 1 | anbi2d 630 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ 𝐵 ∈ ℝ))) |
3 | simpl 485 | . . . . . 6 ⊢ ((𝑥 = 𝐵 ∧ 𝑡 ∈ 𝑇) → 𝑥 = 𝐵) | |
4 | 3 | mpteq2dva 5163 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ 𝐵)) |
5 | 4 | eleq1d 2899 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)) |
6 | 2, 5 | imbi12d 347 | . . 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 3569 | . 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 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ℝcr 10538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ral 3145 df-opab 5131 df-mpt 5149 |
This theorem is referenced by: stoweidlem18 42310 stoweidlem19 42311 stoweidlem22 42314 stoweidlem32 42324 stoweidlem36 42328 stoweidlem40 42332 stoweidlem41 42333 stoweidlem55 42347 |
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