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Theorem msrrcl 32250
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 32249 . . . 4 𝑅:𝑃𝑃
43ffvelrni 6669 . . 3 (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃)
54a1i 11 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃))
63ffvelrni 6669 . . 3 (𝑌𝑃 → (𝑅𝑌) ∈ 𝑃)
7 eleq1 2847 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 ↔ (𝑅𝑌) ∈ 𝑃))
86, 7syl5ibr 238 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑃 → (𝑅𝑋) ∈ 𝑃))
93fdmi 6348 . . . . . 6 dom 𝑅 = 𝑃
10 0nelxp 5434 . . . . . . 7 ¬ ∅ ∈ ((V × V) × V)
111mpstssv 32246 . . . . . . . 8 𝑃 ⊆ ((V × V) × V)
1211sseli 3850 . . . . . . 7 (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
1310, 12mto 189 . . . . . 6 ¬ ∅ ∈ 𝑃
149, 13ndmfvrcl 6524 . . . . 5 ((𝑅𝑋) ∈ 𝑃𝑋𝑃)
1514adantl 474 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑋𝑃)
167biimpa 469 . . . . 5 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑅𝑌) ∈ 𝑃)
179, 13ndmfvrcl 6524 . . . . 5 ((𝑅𝑌) ∈ 𝑃𝑌𝑃)
1816, 17syl 17 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑌𝑃)
1915, 182thd 257 . . 3 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑋𝑃𝑌𝑃))
2019ex 405 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 → (𝑋𝑃𝑌𝑃)))
215, 8, 20pm5.21ndd 372 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  Vcvv 3409  c0 4173   × cxp 5398  cfv 6182  mPreStcmpst 32180  mStRedcmsr 32181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-ot 4444  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-1st 7494  df-2nd 7495  df-mpst 32200  df-msr 32201
This theorem is referenced by:  elmthm  32283
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