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Mirrors > Home > MPE Home > Th. List > Mathboxes > mthmi | Structured version Visualization version GIF version |
Description: A statement whose reduct is the reduct of a provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mthmval.r | ⊢ 𝑅 = (mStRed‘𝑇) |
mthmval.j | ⊢ 𝐽 = (mPPSt‘𝑇) |
mthmval.u | ⊢ 𝑈 = (mThm‘𝑇) |
Ref | Expression |
---|---|
mthmi | ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6678 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑅‘𝑥) = (𝑅‘𝑌) ↔ (𝑅‘𝑋) = (𝑅‘𝑌))) | |
2 | 1 | rspcev 3622 | . 2 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌)) |
3 | mthmval.r | . . 3 ⊢ 𝑅 = (mStRed‘𝑇) | |
4 | mthmval.j | . . 3 ⊢ 𝐽 = (mPPSt‘𝑇) | |
5 | mthmval.u | . . 3 ⊢ 𝑈 = (mThm‘𝑇) | |
6 | 3, 4, 5 | elmthm 32823 | . 2 ⊢ (𝑌 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑌)) |
7 | 2, 6 | sylibr 236 | 1 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ‘cfv 6354 mStRedcmsr 32721 mPPStcmpps 32725 mThmcmthm 32726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-ot 4575 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-1st 7688 df-2nd 7689 df-mpst 32740 df-msr 32741 df-mpps 32745 df-mthm 32746 |
This theorem is referenced by: mppsthm 32826 mthmblem 32827 mthmpps 32829 |
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