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Theorem mthmblem 32901
 Description: Lemma for mthmb 32902. (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 2822 . . . . 5 (mPPSt‘𝑇) = (mPPSt‘𝑇)
3 mthmb.u . . . . 5 𝑈 = (mThm‘𝑇)
41, 2, 3mthmval 32896 . . . 4 𝑈 = (𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
54eleq2i 2905 . . 3 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6 eqid 2822 . . . . . 6 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 32863 . . . . 5 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6494 . . . . 5 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . . 4 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6810 . . . 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 278 . 2 (𝑋𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
13 eleq1 2901 . . . 4 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14 ffun 6497 . . . . . . 7 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
157, 14ax-mp 5 . . . . . 6 Fun 𝑅
16 fvelima 6713 . . . . . 6 ((Fun 𝑅 ∧ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
1715, 16mpan 689 . . . . 5 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
181, 2, 3mthmi 32898 . . . . . 6 ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅𝑥) = (𝑅𝑌)) → 𝑌𝑈)
1918rexlimiva 3267 . . . . 5 (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌) → 𝑌𝑈)
2017, 19syl 17 . . . 4 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈)
2113, 20syl6bi 256 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈))
2221adantld 494 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌𝑈))
2312, 22syl5bi 245 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∃wrex 3131  ◡ccnv 5531   “ cima 5535  Fun wfun 6328   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  mPreStcmpst 32794  mStRedcmsr 32795  mPPStcmpps 32799  mThmcmthm 32800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-ot 4548  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-1st 7675  df-2nd 7676  df-mpst 32814  df-msr 32815  df-mpps 32819  df-mthm 32820 This theorem is referenced by:  mthmb  32902
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