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Theorem elmthm 31811
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 31810 . . 3 𝑈 = (𝑅 “ (𝑅𝐽))
54eleq2i 2842 . 2 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅𝐽)))
6 eqid 2771 . . . . 5 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 31777 . . . 4 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6185 . . . 4 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . 3 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6480 . . 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 31809 . . . . 5 𝐽 ⊆ (mPreSt‘𝑇)
13 fvelimab 6395 . . . . 5 ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
149, 12, 13mp2an 672 . . . 4 ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
1514anbi2i 609 . . 3 ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
1612sseli 3748 . . . . . 6 (𝑥𝐽𝑥 ∈ (mPreSt‘𝑇))
176, 1msrrcl 31778 . . . . . 6 ((𝑅𝑥) = (𝑅𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
1816, 17syl5ibcom 235 . . . . 5 (𝑥𝐽 → ((𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
1918rexlimiv 3175 . . . 4 (∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
2019pm4.71ri 550 . . 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 382   = wceq 1631  wcel 2145  wrex 3062  wss 3723  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  mPreStcmpst 31708  mStRedcmsr 31709  mPPStcmpps 31713  mThmcmthm 31714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  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-ov 6796  df-oprab 6797  df-1st 7315  df-2nd 7316  df-mpst 31728  df-msr 31729  df-mpps 31733  df-mthm 31734
This theorem is referenced by:  mthmi  31812  mthmpps  31817
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