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Mirrors > Home > MPE Home > Th. List > Mathboxes > mstapst | Structured version Visualization version GIF version |
Description: A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mstapst.p | ⊢ 𝑃 = (mPreSt‘𝑇) |
mstapst.s | ⊢ 𝑆 = (mStat‘𝑇) |
Ref | Expression |
---|---|
mstapst | ⊢ 𝑆 ⊆ 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | mstapst.s | . . 3 ⊢ 𝑆 = (mStat‘𝑇) | |
3 | 1, 2 | mstaval 34523 | . 2 ⊢ 𝑆 = ran (mStRed‘𝑇) |
4 | mstapst.p | . . . 4 ⊢ 𝑃 = (mPreSt‘𝑇) | |
5 | 4, 1 | msrf 34521 | . . 3 ⊢ (mStRed‘𝑇):𝑃⟶𝑃 |
6 | frn 6721 | . . 3 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → ran (mStRed‘𝑇) ⊆ 𝑃) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ran (mStRed‘𝑇) ⊆ 𝑃 |
8 | 3, 7 | eqsstri 4015 | 1 ⊢ 𝑆 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊆ wss 3947 ran crn 5676 ⟶wf 6536 ‘cfv 6540 mPreStcmpst 34452 mStRedcmsr 34453 mStatcmsta 34454 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7971 df-2nd 7972 df-mpst 34472 df-msr 34473 df-msta 34474 |
This theorem is referenced by: elmsta 34527 mclsssvlem 34541 mclsax 34548 mclsind 34549 mclsppslem 34562 |
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