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Theorem mthmblem 35585
Description: Lemma for mthmb 35586. (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 2737 . . . . 5 (mPPSt‘𝑇) = (mPPSt‘𝑇)
3 mthmb.u . . . . 5 𝑈 = (mThm‘𝑇)
41, 2, 3mthmval 35580 . . . 4 𝑈 = (𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
54eleq2i 2833 . . 3 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6 eqid 2737 . . . . . 6 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 35547 . . . . 5 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6736 . . . . 5 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . . 4 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 7078 . . . 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 275 . 2 (𝑋𝑈 ↔ (𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))))
13 eleq1 2829 . . . 4 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14 ffun 6739 . . . . . . 7 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
157, 14ax-mp 5 . . . . . 6 Fun 𝑅
16 fvelima 6974 . . . . . 6 ((Fun 𝑅 ∧ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
1715, 16mpan 690 . . . . 5 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
181, 2, 3mthmi 35582 . . . . . 6 ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅𝑥) = (𝑅𝑌)) → 𝑌𝑈)
1918rexlimiva 3147 . . . . 5 (∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌) → 𝑌𝑈)
2017, 19syl 17 . . . 4 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈)
2113, 20biimtrdi 253 . . 3 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) → 𝑌𝑈))
2221adantld 490 . 2 ((𝑅𝑋) = (𝑅𝑌) → ((𝑋 ∈ (mPreSt‘𝑇) ∧ (𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇))) → 𝑌𝑈))
2312, 22biimtrid 242 1 ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  ccnv 5684  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  mPreStcmpst 35478  mStRedcmsr 35479  mPPStcmpps 35483  mThmcmthm 35484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-mpst 35498  df-msr 35499  df-mpps 35503  df-mthm 35504
This theorem is referenced by:  mthmb  35586
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