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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mppspst | Structured version Visualization version GIF version | ||
| Description: A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
| Ref | Expression |
|---|---|
| mppsval.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
| mppsval.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
| Ref | Expression |
|---|---|
| mppspst | ⊢ 𝐽 ⊆ 𝑃 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mppsval.p | . . 3 ⊢ 𝑃 = (mPreSt‘𝑇) | |
| 2 | mppsval.j | . . 3 ⊢ 𝐽 = (mPPSt‘𝑇) | |
| 3 | eqid 2737 | . . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇) | |
| 4 | 1, 2, 3 | mppsval 35577 | . 2 ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} |
| 5 | 1, 2, 3 | mppspstlem 35576 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃 |
| 6 | 4, 5 | eqsstri 4030 | 1 ⊢ 𝐽 ⊆ 𝑃 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 〈cotp 4634 ‘cfv 6561 (class class class)co 7431 {coprab 7432 mPreStcmpst 35478 mClscmcls 35482 mPPStcmpps 35483 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpps 35503 |
| This theorem is referenced by: elmthm 35581 mthmpps 35587 mclspps 35589 |
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