Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypelem8 Structured version   Visualization version   GIF version

Theorem ordtypelem8 8593
 Description: Lemma for ordtype 8600. (Contributed by Mario Carneiro, 25-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
ordtypelem8 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
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 ordtypelem8
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . 6 𝐹 = recs(𝐺)
2 ordtypelem.2 . . . . . 6 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 ordtypelem.3 . . . . . 6 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
4 ordtypelem.5 . . . . . 6 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
5 ordtypelem.6 . . . . . 6 𝑂 = OrdIso(𝑅, 𝐴)
6 ordtypelem.7 . . . . . 6 (𝜑𝑅 We 𝐴)
7 ordtypelem.8 . . . . . 6 (𝜑𝑅 Se 𝐴)
81, 2, 3, 4, 5, 6, 7ordtypelem4 8589 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
9 fdm 6210 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
108, 9syl 17 . . . 4 (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹))
11 inss1 3974 . . . . 5 (𝑇 ∩ dom 𝐹) ⊆ 𝑇
121, 2, 3, 4, 5, 6, 7ordtypelem2 8587 . . . . . 6 (𝜑 → Ord 𝑇)
13 ordsson 7152 . . . . . 6 (Ord 𝑇𝑇 ⊆ On)
1412, 13syl 17 . . . . 5 (𝜑𝑇 ⊆ On)
1511, 14syl5ss 3753 . . . 4 (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On)
1610, 15eqsstrd 3778 . . 3 (𝜑 → dom 𝑂 ⊆ On)
17 epweon 7146 . . . 4 E We On
18 weso 5255 . . . 4 ( E We On → E Or On)
1917, 18ax-mp 5 . . 3 E Or On
20 soss 5203 . . 3 (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂))
2116, 19, 20mpisyl 21 . 2 (𝜑 → E Or dom 𝑂)
22 frn 6212 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
238, 22syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
24 wess 5251 . . . 4 (ran 𝑂𝐴 → (𝑅 We 𝐴𝑅 We ran 𝑂))
2523, 6, 24sylc 65 . . 3 (𝜑𝑅 We ran 𝑂)
26 weso 5255 . . 3 (𝑅 We ran 𝑂𝑅 Or ran 𝑂)
27 sopo 5202 . . 3 (𝑅 Or ran 𝑂𝑅 Po ran 𝑂)
2825, 26, 273syl 18 . 2 (𝜑𝑅 Po ran 𝑂)
29 ffun 6207 . . . 4 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
308, 29syl 17 . . 3 (𝜑 → Fun 𝑂)
31 funforn 6281 . . 3 (Fun 𝑂𝑂:dom 𝑂onto→ran 𝑂)
3230, 31sylib 208 . 2 (𝜑𝑂:dom 𝑂onto→ran 𝑂)
33 epel 5180 . . . . 5 (𝑎 E 𝑏𝑎𝑏)
341, 2, 3, 4, 5, 6, 7ordtypelem6 8591 . . . . 5 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3533, 34syl5bi 232 . . . 4 ((𝜑𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3635ralrimiva 3102 . . 3 (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
3736ralrimivw 3103 . 2 (𝜑 → ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))
38 soisoi 6739 . 2 ((( E Or dom 𝑂𝑅 Po ran 𝑂) ∧ (𝑂:dom 𝑂onto→ran 𝑂 ∧ ∀𝑎 ∈ dom 𝑂𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂𝑎)𝑅(𝑂𝑏)))) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
3921, 28, 32, 37, 38syl22anc 1478 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1630   ∈ wcel 2137  ∀wral 3048  ∃wrex 3049  {crab 3052  Vcvv 3338   ∩ cin 3712   ⊆ wss 3713   class class class wbr 4802   ↦ cmpt 4879   E cep 5176   Po wpo 5183   Or wor 5184   Se wse 5221   We wwe 5222  dom cdm 5264  ran crn 5265   “ cima 5267  Ord word 5881  Oncon0 5882  Fun wfun 6041  ⟶wf 6043  –onto→wfo 6045  ‘cfv 6047   Isom wiso 6048  ℩crio 6771  recscrecs 7634  OrdIsocoi 8577 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-wrecs 7574  df-recs 7635  df-oi 8578 This theorem is referenced by:  ordtypelem9  8594  ordtypelem10  8595  oiiso2  8599
 Copyright terms: Public domain W3C validator