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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 2740 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | mstapst.s | . . 3 ⊢ 𝑆 = (mStat‘𝑇) | |
3 | 1, 2 | mstaval 35512 | . 2 ⊢ 𝑆 = ran (mStRed‘𝑇) |
4 | mstapst.p | . . . 4 ⊢ 𝑃 = (mPreSt‘𝑇) | |
5 | 4, 1 | msrf 35510 | . . 3 ⊢ (mStRed‘𝑇):𝑃⟶𝑃 |
6 | frn 6754 | . . 3 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → ran (mStRed‘𝑇) ⊆ 𝑃) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ran (mStRed‘𝑇) ⊆ 𝑃 |
8 | 3, 7 | eqsstri 4043 | 1 ⊢ 𝑆 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3976 ran crn 5701 ⟶wf 6569 ‘cfv 6573 mPreStcmpst 35441 mStRedcmsr 35442 mStatcmsta 35443 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-ot 4657 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-1st 8030 df-2nd 8031 df-mpst 35461 df-msr 35462 df-msta 35463 |
This theorem is referenced by: elmsta 35516 mclsssvlem 35530 mclsax 35537 mclsind 35538 mclsppslem 35551 |
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