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Theorem ordtypelem9 8598
Description: Lemma for ordtype 8604. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8602 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 8597 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8593 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10 frn 6214 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
119, 10syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
121, 2, 3, 4, 5, 6, 7ordtypelem2 8591 . . . . . . . . . . . . 13 (𝜑 → Ord 𝑇)
13 ordirr 5902 . . . . . . . . . . . . 13 (Ord 𝑇 → ¬ 𝑇𝑇)
1412, 13syl 17 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑇𝑇)
151tfr1a 7660 . . . . . . . . . . . . . . . 16 (Fun 𝐹 ∧ Lim dom 𝐹)
1615simpri 481 . . . . . . . . . . . . . . 15 Lim dom 𝐹
17 limord 5945 . . . . . . . . . . . . . . 15 (Lim dom 𝐹 → Ord dom 𝐹)
1816, 17ax-mp 5 . . . . . . . . . . . . . 14 Ord dom 𝐹
191, 2, 3, 4, 5, 6, 7ordtypelem1 8590 . . . . . . . . . . . . . . . 16 (𝜑𝑂 = (𝐹𝑇))
20 ordtypelem9.1 . . . . . . . . . . . . . . . 16 (𝜑𝑂 ∈ V)
2119, 20eqeltrrd 2840 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 7662 . . . . . . . . . . . . . . . 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 5908 . . . . . . . . . . . . . 14 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2618, 24, 25sylancr 698 . . . . . . . . . . . . 13 (𝜑𝑇 ∈ On)
27 imaeq2 5620 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3283 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3190 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 4807 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralv 3310 . . . . . . . . . . . . . . . . . . . 20 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 3124 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexv 3311 . . . . . . . . . . . . . . . . . 18 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 5620 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3283 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3190 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38syl5bb 272 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3339 . . . . . . . . . . . . . . . 16 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2782 . . . . . . . . . . . . . . 15 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3507 . . . . . . . . . . . . . 14 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 982 . . . . . . . . . . . . 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 3130 . . . . . . . . . . 11 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 3070 . . . . . . . . 9 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4919rneqd 5508 . . . . . . . . . . . . 13 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5279 . . . . . . . . . . . . 13 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50syl6eqr 2812 . . . . . . . . . . . 12 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 472 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3283 . . . . . . . . . 10 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
54 ffun 6209 . . . . . . . . . . . . . 14 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
559, 54syl 17 . . . . . . . . . . . . 13 (𝜑 → Fun 𝑂)
56 funfn 6079 . . . . . . . . . . . . 13 (Fun 𝑂𝑂 Fn dom 𝑂)
5755, 56sylib 208 . . . . . . . . . . . 12 (𝜑𝑂 Fn dom 𝑂)
5857adantr 472 . . . . . . . . . . 11 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
59 breq1 4807 . . . . . . . . . . . 12 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
6059ralrn 6526 . . . . . . . . . . 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 3133 . . . . . . . 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 8596 . . . . . . . . 9 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6766ord 391 . . . . . . . 8 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6867rexlimdva 3169 . . . . . . 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 3750 . . . 4 (𝜑𝐴 ⊆ ran 𝑂)
7211, 71eqssd 3761 . . 3 (𝜑 → ran 𝑂 = 𝐴)
73 isoeq5 6735 . . 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 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cin 3714  wss 3715   class class class wbr 4804  cmpt 4881   E cep 5178   Se wse 5223   We wwe 5224  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Ord word 5883  Oncon0 5884  Lim wlim 5885  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049   Isom wiso 6050  crio 6774  recscrecs 7637  OrdIsocoi 8581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-wrecs 7577  df-recs 7638  df-oi 8582
This theorem is referenced by:  ordtypelem10  8599  ordtype2  8606
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