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Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmsta | Structured version Visualization version GIF version |
Description: A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mthmsta.u | ⊢ 𝑈 = (mThm‘𝑇) |
mthmsta.s | ⊢ 𝑆 = (mPreSt‘𝑇) |
Ref | Expression |
---|---|
mthmsta | ⊢ 𝑈 ⊆ 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | eqid 2734 | . . 3 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇) | |
3 | mthmsta.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 35559 | . 2 ⊢ 𝑈 = (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) |
5 | cnvimass 6101 | . . 3 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ dom (mStRed‘𝑇) | |
6 | mthmsta.s | . . . . 5 ⊢ 𝑆 = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 35526 | . . . 4 ⊢ (mStRed‘𝑇):𝑆⟶𝑆 |
8 | 7 | fdmi 6747 | . . 3 ⊢ dom (mStRed‘𝑇) = 𝑆 |
9 | 5, 8 | sseqtri 4031 | . 2 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ 𝑆 |
10 | 4, 9 | eqsstri 4029 | 1 ⊢ 𝑈 ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊆ wss 3962 ◡ccnv 5687 dom cdm 5688 “ cima 5691 ‘cfv 6562 mPreStcmpst 35457 mStRedcmsr 35458 mPPStcmpps 35462 mThmcmthm 35463 |
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-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
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-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4912 df-iun 4997 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-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-1st 8012 df-2nd 8013 df-mpst 35477 df-msr 35478 df-mthm 35483 |
This theorem is referenced by: mthmpps 35566 |
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