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Theorem msrrcl 31145
Description: If 𝑋 and 𝑌 have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
msrf.r 𝑅 = (mStRed‘𝑇)
Assertion
Ref Expression
msrrcl ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))

Proof of Theorem msrrcl
StepHypRef Expression
1 mpstssv.p . . . . 5 𝑃 = (mPreSt‘𝑇)
2 msrf.r . . . . 5 𝑅 = (mStRed‘𝑇)
31, 2msrf 31144 . . . 4 𝑅:𝑃𝑃
43ffvelrni 6314 . . 3 (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃)
54a1i 11 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃))
63ffvelrni 6314 . . 3 (𝑌𝑃 → (𝑅𝑌) ∈ 𝑃)
7 eleq1 2686 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 ↔ (𝑅𝑌) ∈ 𝑃))
86, 7syl5ibr 236 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑃 → (𝑅𝑋) ∈ 𝑃))
93fdmi 6009 . . . . . 6 dom 𝑅 = 𝑃
10 0nelxp 5103 . . . . . . 7 ¬ ∅ ∈ ((V × V) × V)
111mpstssv 31141 . . . . . . . 8 𝑃 ⊆ ((V × V) × V)
1211sseli 3579 . . . . . . 7 (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
1310, 12mto 188 . . . . . 6 ¬ ∅ ∈ 𝑃
149, 13ndmfvrcl 6176 . . . . 5 ((𝑅𝑋) ∈ 𝑃𝑋𝑃)
1514adantl 482 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑋𝑃)
167biimpa 501 . . . . 5 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑅𝑌) ∈ 𝑃)
179, 13ndmfvrcl 6176 . . . . 5 ((𝑅𝑌) ∈ 𝑃𝑌𝑃)
1816, 17syl 17 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑌𝑃)
1915, 182thd 255 . . 3 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑋𝑃𝑌𝑃))
2019ex 450 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 → (𝑋𝑃𝑌𝑃)))
215, 8, 20pm5.21ndd 369 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891   × cxp 5072  cfv 5847  mPreStcmpst 31075  mStRedcmsr 31076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-ot 4157  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-mpst 31095  df-msr 31096
This theorem is referenced by:  elmthm  31178
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