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Theorem elmthm 32076
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 32075 . . 3 𝑈 = (𝑅 “ (𝑅𝐽))
54eleq2i 2851 . 2 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅𝐽)))
6 eqid 2778 . . . . 5 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 32042 . . . 4 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6293 . . . 4 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . 3 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6602 . . 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 32074 . . . . 5 𝐽 ⊆ (mPreSt‘𝑇)
13 fvelimab 6515 . . . . 5 ((𝑅 Fn (mPreSt‘𝑇) ∧ 𝐽 ⊆ (mPreSt‘𝑇)) → ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
149, 12, 13mp2an 682 . . . 4 ((𝑅𝑋) ∈ (𝑅𝐽) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
1514anbi2i 616 . . 3 ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
1612sseli 3817 . . . . . 6 (𝑥𝐽𝑥 ∈ (mPreSt‘𝑇))
176, 1msrrcl 32043 . . . . . 6 ((𝑅𝑥) = (𝑅𝑋) → (𝑥 ∈ (mPreSt‘𝑇) ↔ 𝑋 ∈ (mPreSt‘𝑇)))
1816, 17syl5ibcom 237 . . . . 5 (𝑥𝐽 → ((𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇)))
1918rexlimiv 3209 . . . 4 (∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋) → 𝑋 ∈ (mPreSt‘𝑇))
2019pm4.71ri 556 . . 3 (∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋)))
2115, 20bitr4i 270 . 2 ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅𝐽)) ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
225, 11, 213bitri 289 1 (𝑋𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  wrex 3091  wss 3792  ccnv 5356  cima 5360   Fn wfn 6132  wf 6133  cfv 6137  mPreStcmpst 31973  mStRedcmsr 31974  mPPStcmpps 31978  mThmcmthm 31979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-ot 4407  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-1st 7447  df-2nd 7448  df-mpst 31993  df-msr 31994  df-mpps 31998  df-mthm 31999
This theorem is referenced by:  mthmi  32077  mthmpps  32082
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