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Theorem msrrcl 31772
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 31771 . . . 4 𝑅:𝑃𝑃
43ffvelrni 6501 . . 3 (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃)
54a1i 11 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃))
63ffvelrni 6501 . . 3 (𝑌𝑃 → (𝑅𝑌) ∈ 𝑃)
7 eleq1 2837 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 ↔ (𝑅𝑌) ∈ 𝑃))
86, 7syl5ibr 236 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑃 → (𝑅𝑋) ∈ 𝑃))
93fdmi 6192 . . . . . 6 dom 𝑅 = 𝑃
10 0nelxp 5283 . . . . . . 7 ¬ ∅ ∈ ((V × V) × V)
111mpstssv 31768 . . . . . . . 8 𝑃 ⊆ ((V × V) × V)
1211sseli 3746 . . . . . . 7 (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
1310, 12mto 188 . . . . . 6 ¬ ∅ ∈ 𝑃
149, 13ndmfvrcl 6360 . . . . 5 ((𝑅𝑋) ∈ 𝑃𝑋𝑃)
1514adantl 467 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑋𝑃)
167biimpa 462 . . . . 5 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑅𝑌) ∈ 𝑃)
179, 13ndmfvrcl 6360 . . . . 5 ((𝑅𝑌) ∈ 𝑃𝑌𝑃)
1816, 17syl 17 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑌𝑃)
1915, 182thd 255 . . 3 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑋𝑃𝑌𝑃))
2019ex 397 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 → (𝑋𝑃𝑌𝑃)))
215, 8, 20pm5.21ndd 368 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061   × cxp 5247  cfv 6031  mPreStcmpst 31702  mStRedcmsr 31703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-ot 4323  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7314  df-2nd 7315  df-mpst 31722  df-msr 31723
This theorem is referenced by:  elmthm  31805
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