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Theorem ordtypelem4 8831
Description: Lemma for ordtype 8842. (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 7882 . . . . . . 7 (Fun 𝐹 ∧ Lim dom 𝐹)
32simpli 484 . . . . . 6 Fun 𝐹
4 funres 6267 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝑇))
53, 4mp1i 13 . . . . 5 (𝜑 → Fun (𝐹𝑇))
65funfnd 6256 . . . 4 (𝜑 → (𝐹𝑇) Fn dom (𝐹𝑇))
7 dmres 5756 . . . . 5 dom (𝐹𝑇) = (𝑇 ∩ dom 𝐹)
87fneq2i 6321 . . . 4 ((𝐹𝑇) Fn dom (𝐹𝑇) ↔ (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
96, 8sylib 219 . . 3 (𝜑 → (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
10 simpr 485 . . . . . . 7 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
1110elin1d 4096 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎𝑇)
1211fvresd 6558 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
13 ssrab2 3977 . . . . . . 7 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤}
14 ssrab2 3977 . . . . . . 7 {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ⊆ 𝐴
1513, 14sstri 3898 . . . . . 6 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ 𝐴
16 ordtypelem.2 . . . . . . 7 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
17 ordtypelem.3 . . . . . . 7 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
18 ordtypelem.5 . . . . . . 7 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
19 ordtypelem.6 . . . . . . 7 𝑂 = OrdIso(𝑅, 𝐴)
20 ordtypelem.7 . . . . . . 7 (𝜑𝑅 We 𝐴)
21 ordtypelem.8 . . . . . . 7 (𝜑𝑅 Se 𝐴)
221, 16, 17, 18, 19, 20, 21ordtypelem3 8830 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
2315, 22sseldi 3887 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ 𝐴)
2412, 23eqeltrd 2883 . . . 4 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) ∈ 𝐴)
2524ralrimiva 3149 . . 3 (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴)
26 ffnfv 6745 . . 3 ((𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ ((𝐹𝑇) Fn (𝑇 ∩ dom 𝐹) ∧ ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴))
279, 25, 26sylanbrc 583 . 2 (𝜑 → (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)
281, 16, 17, 18, 19, 20, 21ordtypelem1 8828 . . 3 (𝜑𝑂 = (𝐹𝑇))
2928feq1d 6367 . 2 (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴))
3027, 29mpbird 258 1 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  cin 3858   class class class wbr 4962  cmpt 5041   Se wse 5400   We wwe 5401  dom cdm 5443  ran crn 5444  cres 5445  cima 5446  Oncon0 6066  Lim wlim 6067  Fun wfun 6219   Fn wfn 6220  wf 6221  cfv 6225  crio 6976  recscrecs 7859  OrdIsocoi 8819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-wrecs 7798  df-recs 7860  df-oi 8820
This theorem is referenced by:  ordtypelem5  8832  ordtypelem6  8833  ordtypelem7  8834  ordtypelem8  8835  ordtypelem9  8836  ordtypelem10  8837
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