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Theorem ordtypelem4 9558
Description: Lemma for ordtype 9569. (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 8432 . . . . . . 7 (Fun 𝐹 ∧ Lim dom 𝐹)
32simpli 483 . . . . . 6 Fun 𝐹
4 funres 6609 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝑇))
53, 4mp1i 13 . . . . 5 (𝜑 → Fun (𝐹𝑇))
65funfnd 6598 . . . 4 (𝜑 → (𝐹𝑇) Fn dom (𝐹𝑇))
7 dmres 6031 . . . . 5 dom (𝐹𝑇) = (𝑇 ∩ dom 𝐹)
87fneq2i 6666 . . . 4 ((𝐹𝑇) Fn dom (𝐹𝑇) ↔ (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
96, 8sylib 218 . . 3 (𝜑 → (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
10 simpr 484 . . . . . . 7 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
1110elin1d 4213 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎𝑇)
1211fvresd 6926 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
13 ssrab2 4089 . . . . . . 7 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤}
14 ssrab2 4089 . . . . . . 7 {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ⊆ 𝐴
1513, 14sstri 4004 . . . . . 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 9557 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
2315, 22sselid 3992 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ 𝐴)
2412, 23eqeltrd 2838 . . . 4 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) ∈ 𝐴)
2524ralrimiva 3143 . . 3 (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴)
26 ffnfv 7138 . . 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 9555 . . 3 (𝜑𝑂 = (𝐹𝑇))
2928feq1d 6720 . 2 (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴))
3027, 29mpbird 257 1 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cin 3961   class class class wbr 5147  cmpt 5230   Se wse 5638   We wwe 5639  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Oncon0 6385  Lim wlim 6386  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  crio 7386  recscrecs 8408  OrdIsocoi 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-oi 9547
This theorem is referenced by:  ordtypelem5  9559  ordtypelem6  9560  ordtypelem7  9561  ordtypelem8  9562  ordtypelem9  9563  ordtypelem10  9564
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