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Theorem ordtypelem9 9036
 Description: Lemma for ordtype 9042. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9040 implies that either ran 𝑂 ⊆ 𝐴 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 9035 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 9031 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6510 . . . 4 (𝜑 → ran 𝑂𝐴)
111, 2, 3, 4, 5, 6, 7ordtypelem2 9029 . . . . . . . . . . 11 (𝜑 → Ord 𝑇)
12 ordirr 6192 . . . . . . . . . . 11 (Ord 𝑇 → ¬ 𝑇𝑇)
1311, 12syl 17 . . . . . . . . . 10 (𝜑 → ¬ 𝑇𝑇)
141tfr1a 8046 . . . . . . . . . . . . . 14 (Fun 𝐹 ∧ Lim dom 𝐹)
1514simpri 489 . . . . . . . . . . . . 13 Lim dom 𝐹
16 limord 6233 . . . . . . . . . . . . 13 (Lim dom 𝐹 → Ord dom 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . 12 Ord dom 𝐹
181, 2, 3, 4, 5, 6, 7ordtypelem1 9028 . . . . . . . . . . . . . 14 (𝜑𝑂 = (𝐹𝑇))
19 ordtypelem9.1 . . . . . . . . . . . . . . 15 (𝜑𝑂𝑉)
2019elexd 3430 . . . . . . . . . . . . . 14 (𝜑𝑂 ∈ V)
2118, 20eqeltrrd 2853 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑇) ∈ V)
221tfr2b 8048 . . . . . . . . . . . . . 14 (Ord 𝑇 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2311, 22syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑇 ∈ dom 𝐹 ↔ (𝐹𝑇) ∈ V))
2421, 23mpbird 260 . . . . . . . . . . . 12 (𝜑𝑇 ∈ dom 𝐹)
25 ordelon 6198 . . . . . . . . . . . 12 ((Ord dom 𝐹𝑇 ∈ dom 𝐹) → 𝑇 ∈ On)
2617, 24, 25sylancr 590 . . . . . . . . . . 11 (𝜑𝑇 ∈ On)
27 imaeq2 5902 . . . . . . . . . . . . . . 15 (𝑎 = 𝑇 → (𝐹𝑎) = (𝐹𝑇))
2827raleqdv 3329 . . . . . . . . . . . . . 14 (𝑎 = 𝑇 → (∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
2928rexbidv 3221 . . . . . . . . . . . . 13 (𝑎 = 𝑇 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
30 breq1 5039 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑐 → (𝑧𝑅𝑡𝑐𝑅𝑡))
3130cbvralvw 3361 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡)
32 breq2 5040 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑏 → (𝑐𝑅𝑡𝑐𝑅𝑏))
3332ralbidv 3126 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3431, 33syl5bb 286 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏))
3534cbvrexvw 3362 . . . . . . . . . . . . . . . 16 (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏)
36 imaeq2 5902 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
3736raleqdv 3329 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (∀𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3837rexbidv 3221 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (∃𝑏𝐴𝑐 ∈ (𝐹𝑥)𝑐𝑅𝑏 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
3935, 38syl5bb 286 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏))
4039cbvrabv 3404 . . . . . . . . . . . . . 14 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
414, 40eqtri 2781 . . . . . . . . . . . . 13 𝑇 = {𝑎 ∈ On ∣ ∃𝑏𝐴𝑐 ∈ (𝐹𝑎)𝑐𝑅𝑏}
4229, 41elrab2 3607 . . . . . . . . . . . 12 (𝑇𝑇 ↔ (𝑇 ∈ On ∧ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4342baib 539 . . . . . . . . . . 11 (𝑇 ∈ On → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4426, 43syl 17 . . . . . . . . . 10 (𝜑 → (𝑇𝑇 ↔ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
4513, 44mtbid 327 . . . . . . . . 9 (𝜑 → ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
46 ralnex 3163 . . . . . . . . 9 (∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ¬ ∃𝑏𝐴𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4745, 46sylibr 237 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4847r19.21bi 3137 . . . . . . 7 ((𝜑𝑏𝐴) → ¬ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏)
4918rneqd 5784 . . . . . . . . . . 11 (𝜑 → ran 𝑂 = ran (𝐹𝑇))
50 df-ima 5541 . . . . . . . . . . 11 (𝐹𝑇) = ran (𝐹𝑇)
5149, 50eqtr4di 2811 . . . . . . . . . 10 (𝜑 → ran 𝑂 = (𝐹𝑇))
5251adantr 484 . . . . . . . . 9 ((𝜑𝑏𝐴) → ran 𝑂 = (𝐹𝑇))
5352raleqdv 3329 . . . . . . . 8 ((𝜑𝑏𝐴) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏))
549ffund 6507 . . . . . . . . . . 11 (𝜑 → Fun 𝑂)
5554funfnd 6371 . . . . . . . . . 10 (𝜑𝑂 Fn dom 𝑂)
5655adantr 484 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑂 Fn dom 𝑂)
57 breq1 5039 . . . . . . . . . 10 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
5857ralrn 6851 . . . . . . . . 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 284 . . . . . . 7 ((𝜑𝑏𝐴) → (∀𝑐 ∈ (𝐹𝑇)𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
6148, 60mtbid 327 . . . . . 6 ((𝜑𝑏𝐴) → ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
62 rexnal 3165 . . . . . 6 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
6361, 62sylibr 237 . . . . 5 ((𝜑𝑏𝐴) → ∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏)
641, 2, 3, 4, 5, 6, 7ordtypelem7 9034 . . . . . . 7 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6564ord 861 . . . . . 6 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6665rexlimdva 3208 . . . . 5 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
6763, 66mpd 15 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
6810, 67eqelssd 3915 . . 3 (𝜑 → ran 𝑂 = 𝐴)
69 isoeq5 7074 . . 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 235 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∩ cin 3859   class class class wbr 5036   ↦ cmpt 5116   E cep 5438   Se wse 5485   We wwe 5486  dom cdm 5528  ran crn 5529   ↾ cres 5530   “ cima 5531  Ord word 6173  Oncon0 6174  Lim wlim 6175  Fun wfun 6334   Fn wfn 6335  ‘cfv 6340   Isom wiso 6341  ℩crio 7113  recscrecs 8023  OrdIsocoi 9019 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-wrecs 7963  df-recs 8024  df-oi 9020 This theorem is referenced by:  ordtypelem10  9037  ordtype2  9044
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