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Mirrors > Home > MPE Home > Th. List > Mathboxes > mppspstlem | Structured version Visualization version GIF version |
Description: Lemma for mppspst 32823. (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 7162 | . 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 4578 | . . . . . . . . . 10 ⊢ 〈𝑑, ℎ, 𝑎〉 = 〈〈𝑑, ℎ〉, 𝑎〉 | |
3 | 2 | eqeq2i 2836 | . . . . . . . . 9 ⊢ (𝑥 = 〈𝑑, ℎ, 𝑎〉 ↔ 𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉) |
4 | 3 | biimpri 230 | . . . . . . . 8 ⊢ (𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 → 𝑥 = 〈𝑑, ℎ, 𝑎〉) |
5 | 4 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 → (𝑥 ∈ 𝑃 ↔ 〈𝑑, ℎ, 𝑎〉 ∈ 𝑃)) |
6 | 5 | biimpar 480 | . . . . . 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 1931 | . . . 4 ⊢ (∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃) |
9 | 8 | exlimivv 1933 | . . 3 ⊢ (∃𝑑∃ℎ∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃) |
10 | 9 | abssi 4048 | . 2 ⊢ {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = 〈〈𝑑, ℎ〉, 𝑎〉 ∧ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ⊆ 𝑃 |
11 | 1, 10 | eqsstri 4003 | 1 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ⊆ wss 3938 〈cop 4575 〈cotp 4577 ‘cfv 6357 (class class class)co 7158 {coprab 7159 mPreStcmpst 32722 mClscmcls 32726 mPPStcmpps 32727 |
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-11 2161 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-nfc 2965 df-in 3945 df-ss 3954 df-ot 4578 df-oprab 7162 |
This theorem is referenced by: mppsval 32821 mppspst 32823 |
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