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Theorem ordtypelem9 8978
Description: Lemma for ordtype 8984. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8982 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 8977 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8973 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6514 . . . 4 (𝜑 → ran 𝑂𝐴)
111, 2, 3, 4, 5, 6, 7ordtypelem2 8971 . . . . . . . . . . 11 (𝜑 → Ord 𝑇)
12 ordirr 6202 . . . . . . . . . . 11 (Ord 𝑇 → ¬ 𝑇𝑇)
1311, 12syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝑇𝑇)
141tfr1a 8019 . . . . . . . . . . . . . 14 (Fun 𝐹 ∧ Lim dom 𝐹)
1514simpri 486 . . . . . . . . . . . . 13 Lim dom 𝐹
16 limord 6243 . . . . . . . . . . . . 13 (Lim dom 𝐹 → Ord dom 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . 12 Ord dom 𝐹
181, 2, 3, 4, 5, 6, 7ordtypelem1 8970 . . . . . . . . . . . . . 14 (𝜑𝑂 = (𝐹𝑇))
19 ordtypelem9.1 . . . . . . . . . . . . . 14 (𝜑𝑂 ∈ V)
2018, 19eqeltrrd 2911 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑇) ∈ V)
211tfr2b 8021 . . . . . . . . . . . . . 14 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2211, 21syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2320, 22mpbird 258 . . . . . . . . . . . 12 (𝜑𝑇 ∈ dom 𝐹)
24 ordelon 6208 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2517, 23, 24sylancr 587 . . . . . . . . . . 11 (𝜑𝑇 ∈ On)
26 imaeq2 5918 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2726raleqdv 3413 . . . . . . . . . . . . . 14 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2827rexbidv 3294 . . . . . . . . . . . . 13 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
29 breq1 5060 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3029cbvralvw 3447 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
31 breq2 5061 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3231ralbidv 3194 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3330, 32syl5bb 284 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3433cbvrexvw 3448 . . . . . . . . . . . . . . . 16 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
35 imaeq2 5918 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3635raleqdv 3413 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3736rexbidv 3294 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3834, 37syl5bb 284 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3938cbvrabv 3489 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
404, 39eqtri 2841 . . . . . . . . . . . . 13 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4128, 40elrab2 3680 . . . . . . . . . . . 12 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4241baib 536 . . . . . . . . . . 11 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4325, 42syl 17 . . . . . . . . . 10 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4413, 43mtbid 325 . . . . . . . . 9 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
45 ralnex 3233 . . . . . . . . 9 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4644, 45sylibr 235 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4746r19.21bi 3205 . . . . . . 7 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4818rneqd 5801 . . . . . . . . . . 11 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
49 df-ima 5561 . . . . . . . . . . 11 (𝐹𝑇) = ran (𝐹𝑇)
5048, 49syl6eqr 2871 . . . . . . . . . 10 (𝜑 → ran 𝑂 = (𝐹𝑇))
5150adantr 481 . . . . . . . . 9 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5251raleqdv 3413 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
539ffund 6511 . . . . . . . . . . 11 (𝜑 → Fun 𝑂)
5453funfnd 6379 . . . . . . . . . 10 (𝜑𝑂 Fn dom 𝑂)
5554adantr 481 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
56 breq1 5060 . . . . . . . . . 10 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
5756ralrn 6846 . . . . . . . . 9 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
5855, 57syl 17 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
5952, 58bitr3d 282 . . . . . . 7 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6047, 59mtbid 325 . . . . . 6 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
61 rexnal 3235 . . . . . 6 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6260, 61sylibr 235 . . . . 5 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
631, 2, 3, 4, 5, 6, 7ordtypelem7 8976 . . . . . . 7 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6463ord 858 . . . . . 6 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6564rexlimdva 3281 . . . . 5 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6662, 65mpd 15 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
6710, 66eqelssd 3985 . . 3 (𝜑 → ran 𝑂 = 𝐴)
68 isoeq5 7063 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
6967, 68syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
708, 69mpbid 233 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cin 3932   class class class wbr 5057  cmpt 5137   E cep 5457   Se wse 5505   We wwe 5506  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Ord word 6183  Oncon0 6184  Lim wlim 6185  Fun wfun 6342   Fn wfn 6343  cfv 6348   Isom wiso 6349  crio 7102  recscrecs 7996  OrdIsocoi 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-wrecs 7936  df-recs 7997  df-oi 8962
This theorem is referenced by:  ordtypelem10  8979  ordtype2  8986
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