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Mirrors > Home > MPE Home > Th. List > Mathboxes > mppsthm | Structured version Visualization version GIF version |
Description: A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mppsthm.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
mppsthm.u | ⊢ 𝑈 = (mThm‘𝑇) |
Ref | Expression |
---|---|
mppsthm | ⊢ 𝐽 ⊆ 𝑈 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥) | |
2 | eqid 2734 | . . . 4 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
3 | mppsthm.j | . . . 4 ⊢ 𝐽 = (mPPSt‘𝑇) | |
4 | mppsthm.u | . . . 4 ⊢ 𝑈 = (mThm‘𝑇) | |
5 | 2, 3, 4 | mthmi 35561 | . . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)) → 𝑥 ∈ 𝑈) |
6 | 1, 5 | mpan2 691 | . 2 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝑈) |
7 | 6 | ssriv 3998 | 1 ⊢ 𝐽 ⊆ 𝑈 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ‘cfv 6562 mStRedcmsr 35458 mPPStcmpps 35462 mThmcmthm 35463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-ot 4639 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-1st 8012 df-2nd 8013 df-mpst 35477 df-msr 35478 df-mpps 35482 df-mthm 35483 |
This theorem is referenced by: (None) |
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