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Theorem mthmblem 31185
Description: Lemma for mthmb 31186. (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 2621 . . . . 5 (mPPSt‘𝑇) = (mPPSt‘𝑇)
3 mthmb.u . . . . 5 𝑈 = (mThm‘𝑇)
41, 2, 3mthmval 31180 . . . 4 𝑈 = (𝑅 “ (𝑅 “ (mPPSt‘𝑇)))
54eleq2i 2690 . . 3 (𝑋𝑈𝑋 ∈ (𝑅 “ (𝑅 “ (mPPSt‘𝑇))))
6 eqid 2621 . . . . . 6 (mPreSt‘𝑇) = (mPreSt‘𝑇)
76, 1msrf 31147 . . . . 5 𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇)
8 ffn 6002 . . . . 5 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → 𝑅 Fn (mPreSt‘𝑇))
97, 8ax-mp 5 . . . 4 𝑅 Fn (mPreSt‘𝑇)
10 elpreima 6293 . . . 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 2686 . . . 4 ((𝑅𝑋) = (𝑅𝑌) → ((𝑅𝑋) ∈ (𝑅 “ (mPPSt‘𝑇)) ↔ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))))
14 ffun 6005 . . . . . . 7 (𝑅:(mPreSt‘𝑇)⟶(mPreSt‘𝑇) → Fun 𝑅)
157, 14ax-mp 5 . . . . . 6 Fun 𝑅
16 fvelima 6205 . . . . . 6 ((Fun 𝑅 ∧ (𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇))) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
1715, 16mpan 705 . . . . 5 ((𝑅𝑌) ∈ (𝑅 “ (mPPSt‘𝑇)) → ∃𝑥 ∈ (mPPSt‘𝑇)(𝑅𝑥) = (𝑅𝑌))
181, 2, 3mthmi 31182 . . . . . 6 ((𝑥 ∈ (mPPSt‘𝑇) ∧ (𝑅𝑥) = (𝑅𝑌)) → 𝑌𝑈)
1918rexlimiva 3021 . . . . 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 483 . 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 384   = wceq 1480  wcel 1987  wrex 2908  ccnv 5073  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  mPreStcmpst 31078  mStRedcmsr 31079  mPPStcmpps 31083  mThmcmthm 31084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-ot 4157  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-1st 7113  df-2nd 7114  df-mpst 31098  df-msr 31099  df-mpps 31103  df-mthm 31104
This theorem is referenced by:  mthmb  31186
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