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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 2823 | . . 3 ⊢ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥) | |
2 | eqid 2823 | . . . 4 ⊢ (mStRed‘𝑇) = (mStRed‘𝑇) | |
3 | mppsthm.j | . . . 4 ⊢ 𝐽 = (mPPSt‘𝑇) | |
4 | mppsthm.u | . . . 4 ⊢ 𝑈 = (mThm‘𝑇) | |
5 | 2, 3, 4 | mthmi 32826 | . . 3 ⊢ ((𝑥 ∈ 𝐽 ∧ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)) → 𝑥 ∈ 𝑈) |
6 | 1, 5 | mpan2 689 | . 2 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝑈) |
7 | 6 | ssriv 3973 | 1 ⊢ 𝐽 ⊆ 𝑈 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 mStRedcmsr 32723 mPPStcmpps 32727 mThmcmthm 32728 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-1st 7691 df-2nd 7692 df-mpst 32742 df-msr 32743 df-mpps 32747 df-mthm 32748 |
This theorem is referenced by: (None) |
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