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Theorem ordtypelem4 8467
 Description: Lemma for ordtype 8478. (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
ordtypelem4 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶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 ordtypelem4
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . . . 8 𝐹 = recs(𝐺)
21tfr1a 7535 . . . . . . 7 (Fun 𝐹 ∧ Lim dom 𝐹)
32simpli 473 . . . . . 6 Fun 𝐹
4 funres 5967 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝑇))
53, 4mp1i 13 . . . . 5 (𝜑 → Fun (𝐹𝑇))
6 funfn 5956 . . . . 5 (Fun (𝐹𝑇) ↔ (𝐹𝑇) Fn dom (𝐹𝑇))
75, 6sylib 208 . . . 4 (𝜑 → (𝐹𝑇) Fn dom (𝐹𝑇))
8 dmres 5454 . . . . 5 dom (𝐹𝑇) = (𝑇 ∩ dom 𝐹)
98fneq2i 6024 . . . 4 ((𝐹𝑇) Fn dom (𝐹𝑇) ↔ (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
107, 9sylib 208 . . 3 (𝜑 → (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
11 inss1 3866 . . . . . . 7 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
12 simpr 476 . . . . . . 7 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
1311, 12sseldi 3634 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎𝑇)
14 fvres 6245 . . . . . 6 (𝑎𝑇 → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
1513, 14syl 17 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
16 ssrab2 3720 . . . . . . 7 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤}
17 ssrab2 3720 . . . . . . 7 {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ⊆ 𝐴
1816, 17sstri 3645 . . . . . 6 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ 𝐴
19 ordtypelem.2 . . . . . . 7 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
20 ordtypelem.3 . . . . . . 7 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
21 ordtypelem.5 . . . . . . 7 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
22 ordtypelem.6 . . . . . . 7 𝑂 = OrdIso(𝑅, 𝐴)
23 ordtypelem.7 . . . . . . 7 (𝜑𝑅 We 𝐴)
24 ordtypelem.8 . . . . . . 7 (𝜑𝑅 Se 𝐴)
251, 19, 20, 21, 22, 23, 24ordtypelem3 8466 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
2618, 25sseldi 3634 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ 𝐴)
2715, 26eqeltrd 2730 . . . 4 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) ∈ 𝐴)
2827ralrimiva 2995 . . 3 (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴)
29 ffnfv 6428 . . 3 ((𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ ((𝐹𝑇) Fn (𝑇 ∩ dom 𝐹) ∧ ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴))
3010, 28, 29sylanbrc 699 . 2 (𝜑 → (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)
311, 19, 20, 21, 22, 23, 24ordtypelem1 8464 . . 3 (𝜑𝑂 = (𝐹𝑇))
3231feq1d 6068 . 2 (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴))
3330, 32mpbird 247 1 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∩ cin 3606   class class class wbr 4685   ↦ cmpt 4762   Se wse 5100   We wwe 5101  dom cdm 5143  ran crn 5144   ↾ cres 5145   “ cima 5146  Oncon0 5761  Lim wlim 5762  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  ℩crio 6650  recscrecs 7512  OrdIsocoi 8455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-wrecs 7452  df-recs 7513  df-oi 8456 This theorem is referenced by:  ordtypelem5  8468  ordtypelem6  8469  ordtypelem7  8470  ordtypelem8  8471  ordtypelem9  8472  ordtypelem10  8473
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