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Theorem msrrcl 35147
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 35146 . . . 4 𝑅:𝑃𝑃
43ffvelcdmi 7087 . . 3 (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃)
54a1i 11 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃))
63ffvelcdmi 7087 . . 3 (𝑌𝑃 → (𝑅𝑌) ∈ 𝑃)
7 eleq1 2817 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 ↔ (𝑅𝑌) ∈ 𝑃))
86, 7imbitrrid 245 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑃 → (𝑅𝑋) ∈ 𝑃))
93fdmi 6728 . . . . . 6 dom 𝑅 = 𝑃
10 0nelxp 5706 . . . . . . 7 ¬ ∅ ∈ ((V × V) × V)
111mpstssv 35143 . . . . . . . 8 𝑃 ⊆ ((V × V) × V)
1211sseli 3974 . . . . . . 7 (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
1310, 12mto 196 . . . . . 6 ¬ ∅ ∈ 𝑃
149, 13ndmfvrcl 6927 . . . . 5 ((𝑅𝑋) ∈ 𝑃𝑋𝑃)
1514adantl 481 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑋𝑃)
167biimpa 476 . . . . 5 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑅𝑌) ∈ 𝑃)
179, 13ndmfvrcl 6927 . . . . 5 ((𝑅𝑌) ∈ 𝑃𝑌𝑃)
1816, 17syl 17 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑌𝑃)
1915, 182thd 265 . . 3 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑋𝑃𝑌𝑃))
2019ex 412 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 → (𝑋𝑃𝑌𝑃)))
215, 8, 20pm5.21ndd 379 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃𝑌𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  Vcvv 3470  c0 4318   × cxp 5670  cfv 6542  mPreStcmpst 35077  mStRedcmsr 35078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-ot 4633  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7987  df-2nd 7988  df-mpst 35097  df-msr 35098
This theorem is referenced by:  elmthm  35180
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