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Mathbox for Mario Carneiro |
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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 2734 | . . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇) | |
4 | 1, 2, 3 | mppsval 35556 | . 2 ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} |
5 | 1, 2, 3 | mppspstlem 35555 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃 |
6 | 4, 5 | eqsstri 4029 | 1 ⊢ 𝐽 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 〈cotp 4638 ‘cfv 6562 (class class class)co 7430 {coprab 7431 mPreStcmpst 35457 mClscmcls 35461 mPPStcmpps 35462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpps 35482 |
This theorem is referenced by: elmthm 35560 mthmpps 35566 mclspps 35568 |
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