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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 2821 | . . 3 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
2 | mstapst.s | . . 3 ⊢ 𝑆 = (mStat‘𝑇) | |
3 | 1, 2 | mstaval 32791 | . 2 ⊢ 𝑆 = ran (mStRed‘𝑇) |
4 | mstapst.p | . . . 4 ⊢ 𝑃 = (mPreSt‘𝑇) | |
5 | 4, 1 | msrf 32789 | . . 3 ⊢ (mStRed‘𝑇):𝑃⟶𝑃 |
6 | frn 6520 | . . 3 ⊢ ((mStRed‘𝑇):𝑃⟶𝑃 → ran (mStRed‘𝑇) ⊆ 𝑃) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ran (mStRed‘𝑇) ⊆ 𝑃 |
8 | 3, 7 | eqsstri 4001 | 1 ⊢ 𝑆 ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3936 ran crn 5556 ⟶wf 6351 ‘cfv 6355 mPreStcmpst 32720 mStRedcmsr 32721 mStatcmsta 32722 |
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-msta 32742 |
This theorem is referenced by: elmsta 32795 mclsssvlem 32809 mclsax 32816 mclsind 32817 mclsppslem 32830 |
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