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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 2622 | . . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇) | |
4 | 1, 2, 3 | mppsval 31469 | . 2 ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} |
5 | 1, 2, 3 | mppspstlem 31468 | . 2 ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃 |
6 | 4, 5 | eqsstri 3635 | 1 ⊢ 𝐽 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 〈cotp 4185 ‘cfv 5888 (class class class)co 6650 {coprab 6651 mPreStcmpst 31370 mClscmcls 31374 mPPStcmpps 31375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpps 31395 |
This theorem is referenced by: elmthm 31473 mthmpps 31479 mclspps 31481 |
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