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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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