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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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