![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mppspstlem | Structured version Visualization version GIF version |
Description: Lemma for mppspst 35415. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mppsval.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
mppsval.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
mppsval.c | ⊢ 𝐶 = (mCls‘𝑇) |
Ref | Expression |
---|---|
mppspstlem | ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oprab 7420 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} = {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} | |
2 | df-ot 4632 | . . . . . . . . . 10 ⊢ 〈𝑑, ℎ, 𝑎〉 = 〈〈𝑑, ℎ〉, 𝑎〉 | |
3 | 2 | eqeq2i 2739 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑑, ℎ, 𝑎〉 ↔ 𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉) |
4 | 3 | biimpri 227 | . . . . . . . 8 ⊢ (𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 → 𝑥 = 〈𝑑, ℎ, 𝑎〉) |
5 | 4 | eleq1d 2811 | . . . . . . 7 ⊢ (𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 → (𝑥 ∈ 𝑃 ↔ 〈𝑑, ℎ, 𝑎〉 ∈ 𝑃)) |
6 | 5 | biimpar 476 | . . . . . 6 ⊢ ((𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ 〈𝑑, ℎ, 𝑎〉 ∈ 𝑃) → 𝑥 ∈ 𝑃) |
7 | 6 | adantrr 715 | . . . . 5 ⊢ ((𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃) |
8 | 7 | exlimiv 1926 | . . . 4 ⊢ (∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃) |
9 | 8 | exlimivv 1928 | . . 3 ⊢ (∃𝑑∃ℎ∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃) |
10 | 9 | abssi 4063 | . 2 ⊢ {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ⊆ 𝑃 |
11 | 1, 10 | eqsstri 4013 | 1 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2703 ⊆ wss 3946 〈cop 4629 〈cotp 4631 ‘cfv 6546 (class class class)co 7416 {coprab 7417 mPreStcmpst 35314 mClscmcls 35318 mPPStcmpps 35319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ss 3963 df-ot 4632 df-oprab 7420 |
This theorem is referenced by: mppsval 35413 mppspst 35415 |
Copyright terms: Public domain | W3C validator |