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Theorem elmthm 31599
Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmval.r 𝑅 = (mStRed‘𝑇)
mthmval.j 𝐽 = (mPPSt‘𝑇)
mthmval.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
elmthm (𝑋𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑅   𝑥,𝑇   𝑥,𝑋
Allowed substitution hint:   𝑈(𝑥)

Proof of Theorem elmthm
StepHypRef Expression
1 mthmval.r . . . 4 𝑅 = (mStRed‘𝑇)
2 mthmval.j . . . 4 𝐽 = (mPPSt‘𝑇)
3 mthmval.u . . . 4 𝑈 = (mThm‘𝑇)
41, 2, 3mthmval 31598 . . 3 𝑈 = (𝑅 “ (𝑅𝐽))
54eleq2i 2722 . 2 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅𝐽)))
6 eqid 2651 . . . . 5 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 31565 . . . 4 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6083 . . . 4 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . 3 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6377 . . 3 (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ (𝑅 “ (𝑅𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽))))
119, 10ax-mp 5 . 2 (𝑋 ∈ (𝑅 “ (𝑅𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)))
126, 2mppspst 31597 . . . . 5 𝐽 ⊆ (mPreSt‘𝑇)
13 fvelimab 6292 . . . . 5 ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
149, 12, 13mp2an 708 . . . 4 ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
1514anbi2i 730 . . 3 ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
1612sseli 3632 . . . . . 6 (𝑥𝐽𝑥 ∈ (mPreSt‘𝑇))
176, 1msrrcl 31566 . . . . . 6 ((𝑅𝑥) = (𝑅𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
1816, 17syl5ibcom 235 . . . . 5 (𝑥𝐽 → ((𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
1918rexlimiv 3056 . . . 4 (∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
2019pm4.71ri 666 . . 3 (∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
2115, 20bitr4i 267 . 2 ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
225, 11, 213bitri 286 1 (𝑋𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  wss 3607  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  mPreStcmpst 31496  mStRedcmsr 31497  mPPStcmpps 31501  mThmcmthm 31502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-1st 7210  df-2nd 7211  df-mpst 31516  df-msr 31517  df-mpps 31521  df-mthm 31522
This theorem is referenced by:  mthmi  31600  mthmpps  31605
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