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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 2798 | . . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇) | |
4 | 1, 2, 3 | mppsval 32932 | . 2 ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} |
5 | 1, 2, 3 | mppspstlem 32931 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃 |
6 | 4, 5 | eqsstri 3949 | 1 ⊢ 𝐽 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 〈cotp 4533 ‘cfv 6324 (class class class)co 7135 {coprab 7136 mPreStcmpst 32833 mClscmcls 32837 mPPStcmpps 32838 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpps 32858 |
This theorem is referenced by: elmthm 32936 mthmpps 32942 mclspps 32944 |
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