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Theorem mthmblem 31755
Description: Lemma for mthmb 31756. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmb.r 𝑅 = (mStRed‘𝑇)
mthmb.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mthmblem ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Proof of Theorem mthmblem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mthmb.r . . . . 5 𝑅 = (mStRed‘𝑇)
2 eqid 2748 . . . . 5 (mPPSt‘𝑇) = (mPPSt‘𝑇)
3 mthmb.u . . . . 5 𝑈 = (mThm‘𝑇)
41, 2, 3mthmval 31750 . . . 4 𝑈 = (𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
54eleq2i 2819 . . 3 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6 eqid 2748 . . . . . 6 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 31717 . . . . 5 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6194 . . . . 5 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . . 4 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6488 . . . 4 (𝑅 Fn (mPreSt‘𝑇) → (𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)))))
119, 10ax-mp 5 . . 3 (𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))) ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
125, 11bitri 264 . 2 (𝑋𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
13 eleq1 2815 . . . 4 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14 ffun 6197 . . . . . . 7 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
157, 14ax-mp 5 . . . . . 6 Fun 𝑅
16 fvelima 6398 . . . . . 6 ((Fun 𝑅 ∧ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
1715, 16mpan 708 . . . . 5 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
181, 2, 3mthmi 31752 . . . . . 6 ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅𝑥) = (𝑅𝑌)) → 𝑌𝑈)
1918rexlimiva 3154 . . . . 5 (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌) → 𝑌𝑈)
2017, 19syl 17 . . . 4 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈)
2113, 20syl6bi 243 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈))
2221adantld 484 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌𝑈))
2312, 22syl5bi 232 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wrex 3039  ccnv 5253  cima 5257  Fun wfun 6031   Fn wfn 6032  wf 6033  cfv 6037  mPreStcmpst 31648  mStRedcmsr 31649  mPPStcmpps 31653  mThmcmthm 31654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-ot 4318  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-1st 7321  df-2nd 7322  df-mpst 31668  df-msr 31669  df-mpps 31673  df-mthm 31674
This theorem is referenced by:  mthmb  31756
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