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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 2821 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | eqid 2821 | . . 3 ⊢ (mPPSt‘𝑇) = (mPPSt‘𝑇) | |
3 | mthmsta.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
4 | 1, 2, 3 | mthmval 32817 | . 2 ⊢ 𝑈 = (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) |
5 | cnvimass 5944 | . . 3 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ dom (mStRed‘𝑇) | |
6 | mthmsta.s | . . . . 5 ⊢ 𝑆 = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 32784 | . . . 4 ⊢ (mStRed‘𝑇):𝑆⟶𝑆 |
8 | 7 | fdmi 6519 | . . 3 ⊢ dom (mStRed‘𝑇) = 𝑆 |
9 | 5, 8 | sseqtri 4003 | . 2 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ 𝑆 |
10 | 4, 9 | eqsstri 4001 | 1 ⊢ 𝑈 ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊆ wss 3936 ◡ccnv 5549 dom cdm 5550 “ cima 5553 ‘cfv 6350 mPreStcmpst 32715 mStRedcmsr 32716 mPPStcmpps 32720 mThmcmthm 32721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-1st 7683 df-2nd 7684 df-mpst 32735 df-msr 32736 df-mthm 32741 |
This theorem is referenced by: mthmpps 32824 |
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