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Theorem ordtypelem9 8472
Description: Lemma for ordtype 8478. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8476 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 8471 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8467 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10 frn 6091 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
119, 10syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
121, 2, 3, 4, 5, 6, 7ordtypelem2 8465 . . . . . . . . . . . . 13 (𝜑 → Ord 𝑇)
13 ordirr 5779 . . . . . . . . . . . . 13 (Ord 𝑇 → ¬ 𝑇𝑇)
1412, 13syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑇𝑇)
151tfr1a 7535 . . . . . . . . . . . . . . . 16 (Fun 𝐹 ∧ Lim dom 𝐹)
1615simpri 477 . . . . . . . . . . . . . . 15 Lim dom 𝐹
17 limord 5822 . . . . . . . . . . . . . . 15 (Lim dom 𝐹 → Ord dom 𝐹)
1816, 17ax-mp 5 . . . . . . . . . . . . . 14 Ord dom 𝐹
191, 2, 3, 4, 5, 6, 7ordtypelem1 8464 . . . . . . . . . . . . . . . 16 (𝜑𝑂 = (𝐹𝑇))
20 ordtypelem9.1 . . . . . . . . . . . . . . . 16 (𝜑𝑂 ∈ V)
2119, 20eqeltrrd 2731 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 7537 . . . . . . . . . . . . . . . 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 5785 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2618, 24, 25sylancr 696 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ On)
27 imaeq2 5497 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3174 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3081 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 4688 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralv 3201 . . . . . . . . . . . . . . . . . . . 20 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 4689 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 3015 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexv 3202 . . . . . . . . . . . . . . . . . 18 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 5497 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3174 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3081 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38syl5bb 272 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3230 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2673 . . . . . . . . . . . . . . 15 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3399 . . . . . . . . . . . . . 14 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 964 . . . . . . . . . . . . 13 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4426, 43syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4514, 44mtbid 313 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
46 ralnex 3021 . . . . . . . . . . 11 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4919rneqd 5385 . . . . . . . . . . . . 13 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5156 . . . . . . . . . . . . 13 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50syl6eqr 2703 . . . . . . . . . . . 12 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 480 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3174 . . . . . . . . . 10 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
54 ffun 6086 . . . . . . . . . . . . . 14 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
559, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun 𝑂)
56 funfn 5956 . . . . . . . . . . . . 13 (Fun 𝑂𝑂 Fn dom 𝑂)
5755, 56sylib 208 . . . . . . . . . . . 12 (𝜑𝑂 Fn dom 𝑂)
5857adantr 480 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
59 breq1 4688 . . . . . . . . . . . 12 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
6059ralrn 6402 . . . . . . . . . . 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 313 . . . . . . . 8 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
64 rexnal 3024 . . . . . . . 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 8470 . . . . . . . . 9 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6766ord 391 . . . . . . . 8 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6867rexlimdva 3060 . . . . . . 7 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6965, 68mpd 15 . . . . . 6 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
7069ex 449 . . . . 5 (𝜑 → (𝑏𝐴𝑏 ∈ ran 𝑂))
7170ssrdv 3642 . . . 4 (𝜑𝐴 ⊆ ran 𝑂)
7211, 71eqssd 3653 . . 3 (𝜑 → ran 𝑂 = 𝐴)
73 isoeq5 6611 . . 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 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607   class class class wbr 4685  cmpt 4762   E cep 5057   Se wse 5100   We wwe 5101  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Ord word 5760  Oncon0 5761  Lim wlim 5762  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926   Isom wiso 5927  crio 6650  recscrecs 7512  OrdIsocoi 8455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-oi 8456
This theorem is referenced by:  ordtypelem10  8473  ordtype2  8480
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