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Theorem ordtypelem9 8785
Description: Lemma for ordtype 8791. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8789 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 8784 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8780 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6351 . . . 4 (𝜑 → ran 𝑂𝐴)
111, 2, 3, 4, 5, 6, 7ordtypelem2 8778 . . . . . . . . . . 11 (𝜑 → Ord 𝑇)
12 ordirr 6047 . . . . . . . . . . 11 (Ord 𝑇 → ¬ 𝑇𝑇)
1311, 12syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝑇𝑇)
141tfr1a 7834 . . . . . . . . . . . . . 14 (Fun 𝐹 ∧ Lim dom 𝐹)
1514simpri 478 . . . . . . . . . . . . 13 Lim dom 𝐹
16 limord 6088 . . . . . . . . . . . . 13 (Lim dom 𝐹 → Ord dom 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . 12 Ord dom 𝐹
181, 2, 3, 4, 5, 6, 7ordtypelem1 8777 . . . . . . . . . . . . . 14 (𝜑𝑂 = (𝐹𝑇))
19 ordtypelem9.1 . . . . . . . . . . . . . 14 (𝜑𝑂 ∈ V)
2018, 19eqeltrrd 2867 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑇) ∈ V)
211tfr2b 7836 . . . . . . . . . . . . . 14 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2211, 21syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2320, 22mpbird 249 . . . . . . . . . . . 12 (𝜑𝑇 ∈ dom 𝐹)
24 ordelon 6053 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2517, 23, 24sylancr 578 . . . . . . . . . . 11 (𝜑𝑇 ∈ On)
26 imaeq2 5766 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2726raleqdv 3355 . . . . . . . . . . . . . 14 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2827rexbidv 3242 . . . . . . . . . . . . 13 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
29 breq1 4932 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3029cbvralv 3383 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
31 breq2 4933 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3231ralbidv 3147 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3330, 32syl5bb 275 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3433cbvrexv 3384 . . . . . . . . . . . . . . . 16 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
35 imaeq2 5766 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3635raleqdv 3355 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3736rexbidv 3242 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3834, 37syl5bb 275 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3938cbvrabv 3412 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
404, 39eqtri 2802 . . . . . . . . . . . . 13 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4128, 40elrab2 3599 . . . . . . . . . . . 12 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4241baib 528 . . . . . . . . . . 11 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4325, 42syl 17 . . . . . . . . . 10 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4413, 43mtbid 316 . . . . . . . . 9 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
45 ralnex 3183 . . . . . . . . 9 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4644, 45sylibr 226 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4746r19.21bi 3158 . . . . . . 7 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4818rneqd 5651 . . . . . . . . . . 11 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
49 df-ima 5420 . . . . . . . . . . 11 (𝐹𝑇) = ran (𝐹𝑇)
5048, 49syl6eqr 2832 . . . . . . . . . 10 (𝜑 → ran 𝑂 = (𝐹𝑇))
5150adantr 473 . . . . . . . . 9 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5251raleqdv 3355 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
539ffund 6348 . . . . . . . . . . 11 (𝜑 → Fun 𝑂)
5453funfnd 6219 . . . . . . . . . 10 (𝜑𝑂 Fn dom 𝑂)
5554adantr 473 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
56 breq1 4932 . . . . . . . . . 10 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
5756ralrn 6679 . . . . . . . . 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 273 . . . . . . 7 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6047, 59mtbid 316 . . . . . 6 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
61 rexnal 3185 . . . . . 6 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6260, 61sylibr 226 . . . . 5 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
631, 2, 3, 4, 5, 6, 7ordtypelem7 8783 . . . . . . 7 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6463ord 850 . . . . . 6 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6564rexlimdva 3229 . . . . 5 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6662, 65mpd 15 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
6710, 66eqelssd 3878 . . 3 (𝜑 → ran 𝑂 = 𝐴)
68 isoeq5 6897 . . 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 224 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3088  wrex 3089  {crab 3092  Vcvv 3415  cin 3828   class class class wbr 4929  cmpt 5008   E cep 5316   Se wse 5364   We wwe 5365  dom cdm 5407  ran crn 5408  cres 5409  cima 5410  Ord word 6028  Oncon0 6029  Lim wlim 6030  Fun wfun 6182   Fn wfn 6183  cfv 6188   Isom wiso 6189  crio 6936  recscrecs 7811  OrdIsocoi 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-wrecs 7750  df-recs 7812  df-oi 8769
This theorem is referenced by:  ordtypelem10  8786  ordtype2  8793
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