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Theorem ordtypelem9 9563
Description: Lemma for ordtype 9569. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9567 implies that either ran 𝑂𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) (Revised by AV, 28-Jul-2024.)
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 (𝜑𝑂𝑉)
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)   𝑉(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,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 9562 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 9558 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6744 . . . 4 (𝜑 → ran 𝑂𝐴)
111, 2, 3, 4, 5, 6, 7ordtypelem2 9556 . . . . . . . . . . 11 (𝜑 → Ord 𝑇)
12 ordirr 6403 . . . . . . . . . . 11 (Ord 𝑇 → ¬ 𝑇𝑇)
1311, 12syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝑇𝑇)
141tfr1a 8432 . . . . . . . . . . . . . 14 (Fun 𝐹 ∧ Lim dom 𝐹)
1514simpri 485 . . . . . . . . . . . . 13 Lim dom 𝐹
16 limord 6445 . . . . . . . . . . . . 13 (Lim dom 𝐹 → Ord dom 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . 12 Ord dom 𝐹
181, 2, 3, 4, 5, 6, 7ordtypelem1 9555 . . . . . . . . . . . . . 14 (𝜑𝑂 = (𝐹𝑇))
19 ordtypelem9.1 . . . . . . . . . . . . . . 15 (𝜑𝑂𝑉)
2019elexd 3501 . . . . . . . . . . . . . 14 (𝜑𝑂 ∈ V)
2118, 20eqeltrrd 2839 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 8434 . . . . . . . . . . . . . 14 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2311, 22syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2421, 23mpbird 257 . . . . . . . . . . . 12 (𝜑𝑇 ∈ dom 𝐹)
25 ordelon 6409 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2617, 24, 25sylancr 587 . . . . . . . . . . 11 (𝜑𝑇 ∈ On)
27 imaeq2 6075 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3323 . . . . . . . . . . . . . 14 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3176 . . . . . . . . . . . . 13 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 5150 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralvw 3234 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 5151 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 3175 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33bitrid 283 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexvw 3235 . . . . . . . . . . . . . . . 16 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 6075 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3323 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3176 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38bitrid 283 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3443 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2762 . . . . . . . . . . . . 13 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3697 . . . . . . . . . . . 12 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 535 . . . . . . . . . . 11 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4426, 43syl 17 . . . . . . . . . 10 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4513, 44mtbid 324 . . . . . . . . 9 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
46 ralnex 3069 . . . . . . . . 9 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 234 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 3248 . . . . . . 7 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4918rneqd 5951 . . . . . . . . . . 11 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5701 . . . . . . . . . . 11 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50eqtr4di 2792 . . . . . . . . . 10 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 480 . . . . . . . . 9 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3323 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
549ffund 6740 . . . . . . . . . . 11 (𝜑 → Fun 𝑂)
5554funfnd 6598 . . . . . . . . . 10 (𝜑𝑂 Fn dom 𝑂)
5655adantr 480 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
57 breq1 5150 . . . . . . . . . 10 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
5857ralrn 7107 . . . . . . . . 9 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
5956, 58syl 17 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6053, 59bitr3d 281 . . . . . . 7 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6148, 60mtbid 324 . . . . . 6 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
62 rexnal 3097 . . . . . 6 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6361, 62sylibr 234 . . . . 5 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
641, 2, 3, 4, 5, 6, 7ordtypelem7 9561 . . . . . . 7 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6564ord 864 . . . . . 6 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6665rexlimdva 3152 . . . . 5 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6763, 66mpd 15 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
6810, 67eqelssd 4016 . . 3 (𝜑 → ran 𝑂 = 𝐴)
69 isoeq5 7340 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
7068, 69syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
718, 70mpbid 232 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cin 3961   class class class wbr 5147  cmpt 5230   E cep 5587   Se wse 5638   We wwe 5639  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Ord word 6384  Oncon0 6385  Lim wlim 6386  Fun wfun 6556   Fn wfn 6557  cfv 6562   Isom wiso 6563  crio 7386  recscrecs 8408  OrdIsocoi 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-oi 9547
This theorem is referenced by:  ordtypelem10  9564  ordtype2  9571
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