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Theorem ordtypelem3 8972
Description: Lemma for ordtype 8984. (Contributed by Mario Carneiro, 24-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)
Assertion
Ref Expression
ordtypelem3 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   𝑅,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   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 ordtypelem3
StepHypRef Expression
1 simpr 485 . . . . 5 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
21elin2d 4173 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀 ∈ dom 𝐹)
3 ordtypelem.1 . . . . 5 𝐹 = recs(𝐺)
43tfr2a 8020 . . . 4 (𝑀 ∈ dom 𝐹 → (𝐹𝑀) = (𝐺‘(𝐹𝑀)))
52, 4syl 17 . . 3 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) = (𝐺‘(𝐹𝑀)))
63tfr1a 8019 . . . . . . . . 9 (Fun 𝐹 ∧ Lim dom 𝐹)
76simpri 486 . . . . . . . 8 Lim dom 𝐹
8 limord 6243 . . . . . . . 8 (Lim dom 𝐹 → Ord dom 𝐹)
97, 8ax-mp 5 . . . . . . 7 Ord dom 𝐹
10 ordelord 6206 . . . . . . 7 ((Ord dom 𝐹𝑀 ∈ dom 𝐹) → Ord 𝑀)
119, 2, 10sylancr 587 . . . . . 6 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → Ord 𝑀)
123tfr2b 8021 . . . . . 6 (Ord 𝑀 → (𝑀 ∈ dom 𝐹 ↔ (𝐹𝑀) ∈ V))
1311, 12syl 17 . . . . 5 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝑀 ∈ dom 𝐹 ↔ (𝐹𝑀) ∈ V))
142, 13mpbid 233 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ V)
15 ordtypelem.2 . . . . . . 7 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
16 rneq 5799 . . . . . . . . . 10 ( = (𝐹𝑀) → ran = ran (𝐹𝑀))
17 df-ima 5561 . . . . . . . . . 10 (𝐹𝑀) = ran (𝐹𝑀)
1816, 17syl6eqr 2871 . . . . . . . . 9 ( = (𝐹𝑀) → ran = (𝐹𝑀))
1918raleqdv 3413 . . . . . . . 8 ( = (𝐹𝑀) → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤))
2019rabbidv 3478 . . . . . . 7 ( = (𝐹𝑀) → {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤})
2115, 20syl5eq 2865 . . . . . 6 ( = (𝐹𝑀) → 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤})
2221raleqdv 3413 . . . . . 6 ( = (𝐹𝑀) → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2321, 22riotaeqbidv 7106 . . . . 5 ( = (𝐹𝑀) → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
24 ordtypelem.3 . . . . 5 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
25 riotaex 7107 . . . . 5 (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ V
2623, 24, 25fvmpt 6761 . . . 4 ((𝐹𝑀) ∈ V → (𝐺‘(𝐹𝑀)) = (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
2714, 26syl 17 . . 3 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐺‘(𝐹𝑀)) = (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
285, 27eqtrd 2853 . 2 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) = (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))
29 ordtypelem.7 . . . . 5 (𝜑𝑅 We 𝐴)
3029adantr 481 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 We 𝐴)
31 ordtypelem.8 . . . . 5 (𝜑𝑅 Se 𝐴)
3231adantr 481 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑅 Se 𝐴)
33 ssrab2 4053 . . . . 5 {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ⊆ 𝐴
3433a1i 11 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ⊆ 𝐴)
351elin1d 4172 . . . . . . 7 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → 𝑀𝑇)
36 imaeq2 5918 . . . . . . . . . . 11 (𝑥 = 𝑀 → (𝐹𝑥) = (𝐹𝑀))
3736raleqdv 3413 . . . . . . . . . 10 (𝑥 = 𝑀 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡))
3837rexbidv 3294 . . . . . . . . 9 (𝑥 = 𝑀 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑡𝐴𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡))
39 ordtypelem.5 . . . . . . . . 9 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
4038, 39elrab2 3680 . . . . . . . 8 (𝑀𝑇 ↔ (𝑀 ∈ On ∧ ∃𝑡𝐴𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡))
4140simprbi 497 . . . . . . 7 (𝑀𝑇 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡)
4235, 41syl 17 . . . . . 6 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑡𝐴𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡)
43 breq1 5060 . . . . . . . . 9 (𝑗 = 𝑧 → (𝑗𝑅𝑤𝑧𝑅𝑤))
4443cbvralvw 3447 . . . . . . . 8 (∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑤)
45 breq2 5061 . . . . . . . . 9 (𝑤 = 𝑡 → (𝑧𝑅𝑤𝑧𝑅𝑡))
4645ralbidv 3194 . . . . . . . 8 (𝑤 = 𝑡 → (∀𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡))
4744, 46syl5bb 284 . . . . . . 7 (𝑤 = 𝑡 → (∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤 ↔ ∀𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡))
4847cbvrexvw 3448 . . . . . 6 (∃𝑤𝐴𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤 ↔ ∃𝑡𝐴𝑧 ∈ (𝐹𝑀)𝑧𝑅𝑡)
4942, 48sylibr 235 . . . . 5 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃𝑤𝐴𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤)
50 rabn0 4336 . . . . 5 ({𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ≠ ∅ ↔ ∃𝑤𝐴𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤)
5149, 50sylibr 235 . . . 4 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ≠ ∅)
52 wereu2 5545 . . . 4 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ({𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ⊆ 𝐴 ∧ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ≠ ∅)) → ∃!𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
5330, 32, 34, 51, 52syl22anc 834 . . 3 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → ∃!𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣)
54 riotacl2 7119 . . 3 (∃!𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣 → (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
5553, 54syl 17 . 2 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
5628, 55eqeltrd 2910 1 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  ∃!wreu 3137  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288   class class class wbr 5057  cmpt 5137   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  cfv 6348  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-riota 7103  df-wrecs 7936  df-recs 7997
This theorem is referenced by:  ordtypelem4  8973  ordtypelem6  8975  ordtypelem7  8976
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