![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
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 2736 | . . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇) | |
4 | 1, 2, 3 | mppsval 34106 | . 2 ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} |
5 | 1, 2, 3 | mppspstlem 34105 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃 |
6 | 4, 5 | eqsstri 3978 | 1 ⊢ 𝐽 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 〈cotp 4594 ‘cfv 6496 (class class class)co 7356 {coprab 7357 mPreStcmpst 34007 mClscmcls 34011 mPPStcmpps 34012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-ot 4595 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpps 34032 |
This theorem is referenced by: elmthm 34110 mthmpps 34116 mclspps 34118 |
Copyright terms: Public domain | W3C validator |