Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msrrcl Structured version   Visualization version   GIF version

Theorem msrrcl 31747
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 31746 . . . 4 𝑅:𝑃𝑃
43ffvelrni 6521 . . 3 (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃)
54a1i 11 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑃 → (𝑅𝑋) ∈ 𝑃))
63ffvelrni 6521 . . 3 (𝑌𝑃 → (𝑅𝑌) ∈ 𝑃)
7 eleq1 2827 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ 𝑃 ↔ (𝑅𝑌) ∈ 𝑃))
86, 7syl5ibr 236 . 2 ((𝑅𝑋) = (𝑅𝑌) → (𝑌𝑃 → (𝑅𝑋) ∈ 𝑃))
93fdmi 6213 . . . . . 6 dom 𝑅 = 𝑃
10 0nelxp 5300 . . . . . . 7 ¬ ∅ ∈ ((V × V) × V)
111mpstssv 31743 . . . . . . . 8 𝑃 ⊆ ((V × V) × V)
1211sseli 3740 . . . . . . 7 (∅ ∈ 𝑃 → ∅ ∈ ((V × V) × V))
1310, 12mto 188 . . . . . 6 ¬ ∅ ∈ 𝑃
149, 13ndmfvrcl 6380 . . . . 5 ((𝑅𝑋) ∈ 𝑃𝑋𝑃)
1514adantl 473 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑋𝑃)
167biimpa 502 . . . . 5 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑅𝑌) ∈ 𝑃)
179, 13ndmfvrcl 6380 . . . . 5 ((𝑅𝑌) ∈ 𝑃𝑌𝑃)
1816, 17syl 17 . . . 4 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → 𝑌𝑃)
1915, 182thd 255 . . 3 (((𝑅𝑋) = (𝑅𝑌) ∧ (𝑅𝑋) ∈ 𝑃) → (𝑋𝑃𝑌𝑃))
2019ex 449 . 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 383   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058   × cxp 5264  cfv 6049  mPreStcmpst 31677  mStRedcmsr 31678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7333  df-2nd 7334  df-mpst 31697  df-msr 31698
This theorem is referenced by:  elmthm  31780
  Copyright terms: Public domain W3C validator