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Theorem ordtypelem6 8975
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
ordtypelem6 ((𝜑𝑀 ∈ dom 𝑂) → (𝑁𝑀 → (𝑂𝑁)𝑅(𝑂𝑀)))
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑀   𝑗,𝑁,𝑢,𝑤   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)   𝑁(𝑥,𝑧,𝑣,𝑡,)   𝑂(𝑧,𝑤,,𝑗)

Proof of Theorem ordtypelem6
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . . . 5 (𝑎 = 𝑁 → (𝐹𝑎) = (𝐹𝑁))
21breq1d 5067 . . . 4 (𝑎 = 𝑁 → ((𝐹𝑎)𝑅(𝐹𝑀) ↔ (𝐹𝑁)𝑅(𝐹𝑀)))
3 ssrab2 4053 . . . . . . . 8 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤}
4 simpr 485 . . . . . . . . . 10 ((𝜑𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝑂)
5 ordtypelem.1 . . . . . . . . . . . . 13 𝐹 = recs(𝐺)
6 ordtypelem.2 . . . . . . . . . . . . 13 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
7 ordtypelem.3 . . . . . . . . . . . . 13 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
8 ordtypelem.5 . . . . . . . . . . . . 13 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
9 ordtypelem.6 . . . . . . . . . . . . 13 𝑂 = OrdIso(𝑅, 𝐴)
10 ordtypelem.7 . . . . . . . . . . . . 13 (𝜑𝑅 We 𝐴)
11 ordtypelem.8 . . . . . . . . . . . . 13 (𝜑𝑅 Se 𝐴)
125, 6, 7, 8, 9, 10, 11ordtypelem4 8973 . . . . . . . . . . . 12 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
1312fdmd 6516 . . . . . . . . . . 11 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
1413adantr 481 . . . . . . . . . 10 ((𝜑𝑀 ∈ dom 𝑂) → dom 𝑂 = (𝑇 ∩ dom 𝐹))
154, 14eleqtrd 2912 . . . . . . . . 9 ((𝜑𝑀 ∈ dom 𝑂) → 𝑀 ∈ (𝑇 ∩ dom 𝐹))
165, 6, 7, 8, 9, 10, 11ordtypelem3 8972 . . . . . . . . 9 ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
1715, 16syldan 591 . . . . . . . 8 ((𝜑𝑀 ∈ dom 𝑂) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
183, 17sseldi 3962 . . . . . . 7 ((𝜑𝑀 ∈ dom 𝑂) → (𝐹𝑀) ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤})
19 breq2 5061 . . . . . . . . . 10 (𝑤 = (𝐹𝑀) → (𝑗𝑅𝑤𝑗𝑅(𝐹𝑀)))
2019ralbidv 3194 . . . . . . . . 9 (𝑤 = (𝐹𝑀) → (∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤 ↔ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀)))
2120elrab 3677 . . . . . . . 8 ((𝐹𝑀) ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ↔ ((𝐹𝑀) ∈ 𝐴 ∧ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀)))
2221simprbi 497 . . . . . . 7 ((𝐹𝑀) ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} → ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀))
2318, 22syl 17 . . . . . 6 ((𝜑𝑀 ∈ dom 𝑂) → ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀))
245tfr1a 8019 . . . . . . . . 9 (Fun 𝐹 ∧ Lim dom 𝐹)
2524simpli 484 . . . . . . . 8 Fun 𝐹
26 funfn 6378 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
2725, 26mpbi 231 . . . . . . 7 𝐹 Fn dom 𝐹
2824simpri 486 . . . . . . . . 9 Lim dom 𝐹
29 limord 6243 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
3028, 29ax-mp 5 . . . . . . . 8 Ord dom 𝐹
31 inss2 4203 . . . . . . . . . 10 (𝑇 ∩ dom 𝐹) ⊆ dom 𝐹
3213, 31eqsstrdi 4018 . . . . . . . . 9 (𝜑 → dom 𝑂 ⊆ dom 𝐹)
3332sselda 3964 . . . . . . . 8 ((𝜑𝑀 ∈ dom 𝑂) → 𝑀 ∈ dom 𝐹)
34 ordelss 6200 . . . . . . . 8 ((Ord dom 𝐹𝑀 ∈ dom 𝐹) → 𝑀 ⊆ dom 𝐹)
3530, 33, 34sylancr 587 . . . . . . 7 ((𝜑𝑀 ∈ dom 𝑂) → 𝑀 ⊆ dom 𝐹)
36 breq1 5060 . . . . . . . 8 (𝑗 = (𝐹𝑎) → (𝑗𝑅(𝐹𝑀) ↔ (𝐹𝑎)𝑅(𝐹𝑀)))
3736ralima 6991 . . . . . . 7 ((𝐹 Fn dom 𝐹𝑀 ⊆ dom 𝐹) → (∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀) ↔ ∀𝑎𝑀 (𝐹𝑎)𝑅(𝐹𝑀)))
3827, 35, 37sylancr 587 . . . . . 6 ((𝜑𝑀 ∈ dom 𝑂) → (∀𝑗 ∈ (𝐹𝑀)𝑗𝑅(𝐹𝑀) ↔ ∀𝑎𝑀 (𝐹𝑎)𝑅(𝐹𝑀)))
3923, 38mpbid 233 . . . . 5 ((𝜑𝑀 ∈ dom 𝑂) → ∀𝑎𝑀 (𝐹𝑎)𝑅(𝐹𝑀))
4039adantrr 713 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → ∀𝑎𝑀 (𝐹𝑎)𝑅(𝐹𝑀))
41 simprr 769 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → 𝑁𝑀)
422, 40, 41rspcdva 3622 . . 3 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝐹𝑁)𝑅(𝐹𝑀))
435, 6, 7, 8, 9, 10, 11ordtypelem1 8970 . . . . . 6 (𝜑𝑂 = (𝐹𝑇))
4443adantr 481 . . . . 5 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → 𝑂 = (𝐹𝑇))
4544fveq1d 6665 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝑂𝑁) = ((𝐹𝑇)‘𝑁))
465, 6, 7, 8, 9, 10, 11ordtypelem2 8971 . . . . . . 7 (𝜑 → Ord 𝑇)
47 inss1 4202 . . . . . . . . . 10 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
4813, 47eqsstrdi 4018 . . . . . . . . 9 (𝜑 → dom 𝑂𝑇)
4948sselda 3964 . . . . . . . 8 ((𝜑𝑀 ∈ dom 𝑂) → 𝑀𝑇)
5049adantrr 713 . . . . . . 7 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → 𝑀𝑇)
51 ordelss 6200 . . . . . . 7 ((Ord 𝑇𝑀𝑇) → 𝑀𝑇)
5246, 50, 51syl2an2r 681 . . . . . 6 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → 𝑀𝑇)
5352, 41sseldd 3965 . . . . 5 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → 𝑁𝑇)
5453fvresd 6683 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → ((𝐹𝑇)‘𝑁) = (𝐹𝑁))
5545, 54eqtrd 2853 . . 3 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝑂𝑁) = (𝐹𝑁))
5644fveq1d 6665 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝑂𝑀) = ((𝐹𝑇)‘𝑀))
5750fvresd 6683 . . . 4 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → ((𝐹𝑇)‘𝑀) = (𝐹𝑀))
5856, 57eqtrd 2853 . . 3 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝑂𝑀) = (𝐹𝑀))
5942, 55, 583brtr4d 5089 . 2 ((𝜑 ∧ (𝑀 ∈ dom 𝑂𝑁𝑀)) → (𝑂𝑁)𝑅(𝑂𝑀))
6059expr 457 1 ((𝜑𝑀 ∈ dom 𝑂) → (𝑁𝑀 → (𝑂𝑁)𝑅(𝑂𝑀)))
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  wss 3933   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   Fn wfn 6343  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  df-oi 8962
This theorem is referenced by:  ordtypelem8  8977
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