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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 2734 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | mstapst.s | . . 3 ⊢ 𝑆 = (mStat‘𝑇) | |
3 | 1, 2 | mstaval 35504 | . 2 ⊢ 𝑆 = ran (mStRed‘𝑇) |
4 | mstapst.p | . . . 4 ⊢ 𝑃 = (mPreSt‘𝑇) | |
5 | 4, 1 | msrf 35502 | . . 3 ⊢ (mStRed‘𝑇):𝑃⟶𝑃 |
6 | frn 6753 | . . 3 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → ran (mStRed‘𝑇) ⊆ 𝑃) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ran (mStRed‘𝑇) ⊆ 𝑃 |
8 | 3, 7 | eqsstri 4037 | 1 ⊢ 𝑆 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3970 ran crn 5700 ⟶wf 6568 ‘cfv 6572 mPreStcmpst 35433 mStRedcmsr 35434 mStatcmsta 35435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-1st 8026 df-2nd 8027 df-mpst 35453 df-msr 35454 df-msta 35455 |
This theorem is referenced by: elmsta 35508 mclsssvlem 35522 mclsax 35529 mclsind 35530 mclsppslem 35543 |
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