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Theorem ordtypelem9 8314
 Description: Lemma for ordtype 8320. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8318 implies that either ran 𝑂 ⊆ 𝐴 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 8313 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8309 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10 frn 5966 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
119, 10syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
121, 2, 3, 4, 5, 6, 7ordtypelem2 8307 . . . . . . . . . . . . 13 (𝜑 → Ord 𝑇)
13 ordirr 5658 . . . . . . . . . . . . 13 (Ord 𝑇 → ¬ 𝑇𝑇)
1412, 13syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑇𝑇)
151tfr1a 7377 . . . . . . . . . . . . . . . 16 (Fun 𝐹 ∧ Lim dom 𝐹)
1615simpri 477 . . . . . . . . . . . . . . 15 Lim dom 𝐹
17 limord 5701 . . . . . . . . . . . . . . 15 (Lim dom 𝐹 → Ord dom 𝐹)
1816, 17ax-mp 5 . . . . . . . . . . . . . 14 Ord dom 𝐹
191, 2, 3, 4, 5, 6, 7ordtypelem1 8306 . . . . . . . . . . . . . . . 16 (𝜑𝑂 = (𝐹𝑇))
20 ordtypelem9.1 . . . . . . . . . . . . . . . 16 (𝜑𝑂 ∈ V)
2119, 20eqeltrrd 2689 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 7379 . . . . . . . . . . . . . . . 16 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2312, 22syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2421, 23mpbird 246 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ dom 𝐹)
25 ordelon 5664 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2618, 24, 25sylancr 694 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ On)
27 imaeq2 5381 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3121 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3034 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 4586 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralv 3147 . . . . . . . . . . . . . . . . . . . 20 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 4587 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 2969 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33syl5bb 271 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexv 3148 . . . . . . . . . . . . . . . . . 18 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 5381 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3121 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38syl5bb 271 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3172 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2632 . . . . . . . . . . . . . . 15 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3333 . . . . . . . . . . . . . 14 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 942 . . . . . . . . . . . . 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 2975 . . . . . . . . . . 11 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 223 . . . . . . . . . 10 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 2916 . . . . . . . . 9 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4919rneqd 5274 . . . . . . . . . . . . 13 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5051 . . . . . . . . . . . . 13 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50syl6eqr 2662 . . . . . . . . . . . 12 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 480 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3121 . . . . . . . . . 10 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
54 ffun 5961 . . . . . . . . . . . . . 14 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
559, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun 𝑂)
56 funfn 5833 . . . . . . . . . . . . 13 (Fun 𝑂𝑂 Fn dom 𝑂)
5755, 56sylib 207 . . . . . . . . . . . 12 (𝜑𝑂 Fn dom 𝑂)
5857adantr 480 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
59 breq1 4586 . . . . . . . . . . . 12 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
6059ralrn 6270 . . . . . . . . . . 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 269 . . . . . . . . 9 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6348, 62mtbid 313 . . . . . . . 8 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
64 rexnal 2978 . . . . . . . 8 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6563, 64sylibr 223 . . . . . . 7 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
661, 2, 3, 4, 5, 6, 7ordtypelem7 8312 . . . . . . . . 9 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6766ord 391 . . . . . . . 8 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6867rexlimdva 3013 . . . . . . 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 3574 . . . 4 (𝜑𝐴 ⊆ ran 𝑂)
7211, 71eqssd 3585 . . 3 (𝜑 → ran 𝑂 = 𝐴)
73 isoeq5 6471 . . 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 221 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   Se wse 4995   We wwe 4996  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804   Isom wiso 5805  ℩crio 6510  recscrecs 7354  OrdIsocoi 8297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-oi 8298 This theorem is referenced by:  ordtypelem10  8315  ordtype2  8322
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