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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 2622 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | eqid 2622 | . . 3 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇) | |
3 | mthmsta.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 31472 | . 2 ⊢ 𝑈 = (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) |
5 | cnvimass 5485 | . . 3 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ dom (mStRed‘𝑇) | |
6 | mthmsta.s | . . . . 5 ⊢ 𝑆 = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 31439 | . . . 4 ⊢ (mStRed‘𝑇):𝑆⟶𝑆 |
8 | 7 | fdmi 6052 | . . 3 ⊢ dom (mStRed‘𝑇) = 𝑆 |
9 | 5, 8 | sseqtri 3637 | . 2 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ 𝑆 |
10 | 4, 9 | eqsstri 3635 | 1 ⊢ 𝑈 ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊆ wss 3574 ◡ccnv 5113 dom cdm 5114 “ cima 5117 ‘cfv 5888 mPreStcmpst 31370 mStRedcmsr 31371 mPPStcmpps 31375 mThmcmthm 31376 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 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-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1st 7168 df-2nd 7169 df-mpst 31390 df-msr 31391 df-mthm 31396 |
This theorem is referenced by: mthmpps 31479 |
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