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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 32822 | . 2 ⊢ 𝑈 = (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) |
5 | cnvimass 5949 | . . 3 ⊢ (◡(mStRed‘𝑇) “ ((mStRed‘𝑇) “ (mPPSt‘𝑇))) ⊆ dom (mStRed‘𝑇) | |
6 | mthmsta.s | . . . . 5 ⊢ 𝑆 = (mPreSt‘𝑇) | |
7 | 6, 1 | msrf 32789 | . . . 4 ⊢ (mStRed‘𝑇):𝑆⟶𝑆 |
8 | 7 | fdmi 6524 | . . 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 1537 ⊆ wss 3936 ◡ccnv 5554 dom cdm 5555 “ cima 5558 ‘cfv 6355 mPreStcmpst 32720 mStRedcmsr 32721 mPPStcmpps 32725 mThmcmthm 32726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-1st 7689 df-2nd 7690 df-mpst 32740 df-msr 32741 df-mthm 32746 |
This theorem is referenced by: mthmpps 32829 |
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