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Theorem ordtypelem4 9511
Description: Lemma for ordtype 9522. (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 8388 . . . . . . 7 (Fun 𝐹 ∧ Lim dom 𝐹)
32simpli 485 . . . . . 6 Fun 𝐹
4 funres 6586 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝑇))
53, 4mp1i 13 . . . . 5 (𝜑 → Fun (𝐹𝑇))
65funfnd 6575 . . . 4 (𝜑 → (𝐹𝑇) Fn dom (𝐹𝑇))
7 dmres 6000 . . . . 5 dom (𝐹𝑇) = (𝑇 ∩ dom 𝐹)
87fneq2i 6643 . . . 4 ((𝐹𝑇) Fn dom (𝐹𝑇) ↔ (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
96, 8sylib 217 . . 3 (𝜑 → (𝐹𝑇) Fn (𝑇 ∩ dom 𝐹))
10 simpr 486 . . . . . . 7 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹))
1110elin1d 4196 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎𝑇)
1211fvresd 6907 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) = (𝐹𝑎))
13 ssrab2 4075 . . . . . . 7 {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤}
14 ssrab2 4075 . . . . . . 7 {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ⊆ 𝐴
1513, 14sstri 3989 . . . . . 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 9510 . . . . . 6 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
2315, 22sselid 3978 . . . . 5 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑎) ∈ 𝐴)
2412, 23eqeltrd 2834 . . . 4 ((𝜑𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹𝑇)‘𝑎) ∈ 𝐴)
2524ralrimiva 3147 . . 3 (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴)
26 ffnfv 7112 . . 3 ((𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ ((𝐹𝑇) Fn (𝑇 ∩ dom 𝐹) ∧ ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹𝑇)‘𝑎) ∈ 𝐴))
279, 25, 26sylanbrc 584 . 2 (𝜑 → (𝐹𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)
281, 16, 17, 18, 19, 20, 21ordtypelem1 9508 . . 3 (𝜑𝑂 = (𝐹𝑇))
2928feq1d 6698 . 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 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cin 3945   class class class wbr 5146  cmpt 5229   Se wse 5627   We wwe 5628  dom cdm 5674  ran crn 5675  cres 5676  cima 5677  Oncon0 6360  Lim wlim 6361  Fun wfun 6533   Fn wfn 6534  wf 6535  cfv 6539  crio 7358  recscrecs 8364  OrdIsocoi 9499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-oi 9500
This theorem is referenced by:  ordtypelem5  9512  ordtypelem6  9513  ordtypelem7  9514  ordtypelem8  9515  ordtypelem9  9516  ordtypelem10  9517
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