MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypelem9 Structured version   Visualization version   GIF version

Theorem ordtypelem9 8375
Description: Lemma for ordtype 8381. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8379 implies that either ran 𝑂𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1 𝐹 = recs(𝐺)
ordtypelem.2 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
ordtypelem.3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
ordtypelem.6 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7 (𝜑𝑅 We 𝐴)
ordtypelem.8 (𝜑𝑅 Se 𝐴)
ordtypelem9.1 (𝜑𝑂 ∈ V)
Assertion
Ref Expression
ordtypelem9 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,,𝑗)   𝐶(𝑥,𝑧,𝑤,𝑡,,𝑗)   𝑇(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝐺(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝑂(𝑧,𝑤,,𝑗)

Proof of Theorem ordtypelem9
Dummy variables 𝑎 𝑏 𝑐 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3 𝐹 = recs(𝐺)
2 ordtypelem.2 . . 3 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 ordtypelem.3 . . 3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
4 ordtypelem.5 . . 3 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
5 ordtypelem.6 . . 3 𝑂 = OrdIso(𝑅, 𝐴)
6 ordtypelem.7 . . 3 (𝜑𝑅 We 𝐴)
7 ordtypelem.8 . . 3 (𝜑𝑅 Se 𝐴)
81, 2, 3, 4, 5, 6, 7ordtypelem8 8374 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8370 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10 frn 6010 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
119, 10syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
121, 2, 3, 4, 5, 6, 7ordtypelem2 8368 . . . . . . . . . . . . 13 (𝜑 → Ord 𝑇)
13 ordirr 5700 . . . . . . . . . . . . 13 (Ord 𝑇 → ¬ 𝑇𝑇)
1412, 13syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑇𝑇)
151tfr1a 7435 . . . . . . . . . . . . . . . 16 (Fun 𝐹 ∧ Lim dom 𝐹)
1615simpri 478 . . . . . . . . . . . . . . 15 Lim dom 𝐹
17 limord 5743 . . . . . . . . . . . . . . 15 (Lim dom 𝐹 → Ord dom 𝐹)
1816, 17ax-mp 5 . . . . . . . . . . . . . 14 Ord dom 𝐹
191, 2, 3, 4, 5, 6, 7ordtypelem1 8367 . . . . . . . . . . . . . . . 16 (𝜑𝑂 = (𝐹𝑇))
20 ordtypelem9.1 . . . . . . . . . . . . . . . 16 (𝜑𝑂 ∈ V)
2119, 20eqeltrrd 2699 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 7437 . . . . . . . . . . . . . . . 16 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2312, 22syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2421, 23mpbird 247 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ dom 𝐹)
25 ordelon 5706 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2618, 24, 25sylancr 694 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ On)
27 imaeq2 5421 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3133 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3045 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 4616 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralv 3159 . . . . . . . . . . . . . . . . . . . 20 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 4617 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 2980 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexv 3160 . . . . . . . . . . . . . . . . . 18 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 5421 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3133 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3045 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38syl5bb 272 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3185 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2643 . . . . . . . . . . . . . . 15 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3348 . . . . . . . . . . . . . 14 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 943 . . . . . . . . . . . . 13 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4426, 43syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4514, 44mtbid 314 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
46 ralnex 2986 . . . . . . . . . . 11 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 2927 . . . . . . . . 9 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4919rneqd 5313 . . . . . . . . . . . . 13 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5087 . . . . . . . . . . . . 13 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50syl6eqr 2673 . . . . . . . . . . . 12 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 481 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3133 . . . . . . . . . 10 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
54 ffun 6005 . . . . . . . . . . . . . 14 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
559, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun 𝑂)
56 funfn 5877 . . . . . . . . . . . . 13 (Fun 𝑂𝑂 Fn dom 𝑂)
5755, 56sylib 208 . . . . . . . . . . . 12 (𝜑𝑂 Fn dom 𝑂)
5857adantr 481 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
59 breq1 4616 . . . . . . . . . . . 12 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
6059ralrn 6318 . . . . . . . . . . 11 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6158, 60syl 17 . . . . . . . . . 10 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6253, 61bitr3d 270 . . . . . . . . 9 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6348, 62mtbid 314 . . . . . . . 8 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
64 rexnal 2989 . . . . . . . 8 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6563, 64sylibr 224 . . . . . . 7 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
661, 2, 3, 4, 5, 6, 7ordtypelem7 8373 . . . . . . . . 9 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6766ord 392 . . . . . . . 8 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6867rexlimdva 3024 . . . . . . 7 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6965, 68mpd 15 . . . . . 6 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
7069ex 450 . . . . 5 (𝜑 → (𝑏𝐴𝑏 ∈ ran 𝑂))
7170ssrdv 3589 . . . 4 (𝜑𝐴 ⊆ ran 𝑂)
7211, 71eqssd 3600 . . 3 (𝜑 → ran 𝑂 = 𝐴)
73 isoeq5 6525 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
7472, 73syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
758, 74mpbid 222 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3554  wss 3555   class class class wbr 4613  cmpt 4673   E cep 4983   Se wse 5031   We wwe 5032  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Ord word 5681  Oncon0 5682  Lim wlim 5683  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847   Isom wiso 5848  crio 6564  recscrecs 7412  OrdIsocoi 8358
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-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-3or 1037  df-3an 1038  df-tru 1483  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-rmo 2915  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-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  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-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  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-isom 5856  df-riota 6565  df-wrecs 7352  df-recs 7413  df-oi 8359
This theorem is referenced by:  ordtypelem10  8376  ordtype2  8383
  Copyright terms: Public domain W3C validator