MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypelem10 Structured version   Visualization version   GIF version

Theorem ordtypelem10 8376
Description: Lemma for ordtype 8381. Using ax-rep 4731, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (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
ordtypelem10 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 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 ordtypelem10
Dummy variables 𝑏 𝑐 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3 𝐹 = recs(𝐺)
2 ordtypelem.2 . . 3 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
3 ordtypelem.3 . . 3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
4 ordtypelem.5 . . 3 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
5 ordtypelem.6 . . 3 𝑂 = OrdIso(𝑅, 𝐴)
6 ordtypelem.7 . . 3 (𝜑𝑅 We 𝐴)
7 ordtypelem.8 . . 3 (𝜑𝑅 Se 𝐴)
81, 2, 3, 4, 5, 6, 7ordtypelem8 8374 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8370 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
10 frn 6010 . . . . 5 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂𝐴)
119, 10syl 17 . . . 4 (𝜑 → ran 𝑂𝐴)
12 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏𝐴)
136adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
147adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
151, 2, 3, 4, 5, 13, 14ordtypelem8 8374 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
16 isof1o 6527 . . . . . . . . . . . . 13 (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
17 f1of 6094 . . . . . . . . . . . . 13 (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂⟶ran 𝑂)
1815, 16, 173syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂⟶ran 𝑂)
19 f1of1 6093 . . . . . . . . . . . . . 14 (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1→ran 𝑂)
2015, 16, 193syl 18 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂1-1→ran 𝑂)
21 simpl 473 . . . . . . . . . . . . . . 15 ((𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏𝐴)
22 seex 5037 . . . . . . . . . . . . . . 15 ((𝑅 Se 𝐴𝑏𝐴) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
237, 21, 22syl2an 494 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
2411adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂𝐴)
25 rexnal 2989 . . . . . . . . . . . . . . . . . . 19 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
261, 2, 3, 4, 5, 6, 7ordtypelem7 8373 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2726ord 392 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2827rexlimdva 3024 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2925, 28syl5bir 233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑏𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
3029con1d 139 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3130impr 648 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
32 ffun 6005 . . . . . . . . . . . . . . . . . . . 20 (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂)
339, 32syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → Fun 𝑂)
34 funfn 5877 . . . . . . . . . . . . . . . . . . 19 (Fun 𝑂𝑂 Fn dom 𝑂)
3533, 34sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑𝑂 Fn dom 𝑂)
3635adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
37 breq1 4616 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
3837ralrn 6318 . . . . . . . . . . . . . . . . 17 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3936, 38syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
4031, 39mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
41 ssrab 3659 . . . . . . . . . . . . . . 15 (ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏} ↔ (ran 𝑂𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
4224, 40, 41sylanbrc 697 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏})
4323, 42ssexd 4765 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
44 f1dmex 7083 . . . . . . . . . . . . 13 ((𝑂:dom 𝑂1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
4520, 43, 44syl2anc 692 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
46 fex 6444 . . . . . . . . . . . 12 ((𝑂:dom 𝑂⟶ran 𝑂 ∧ dom 𝑂 ∈ V) → 𝑂 ∈ V)
4718, 45, 46syl2anc 692 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
481, 2, 3, 4, 5, 13, 14, 47ordtypelem9 8375 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
49 isof1o 6527 . . . . . . . . . 10 (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂1-1-onto𝐴)
50 f1ofo 6101 . . . . . . . . . 10 (𝑂:dom 𝑂1-1-onto𝐴𝑂:dom 𝑂onto𝐴)
51 forn 6075 . . . . . . . . . 10 (𝑂:dom 𝑂onto𝐴 → ran 𝑂 = 𝐴)
5248, 49, 50, 514syl 19 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
5312, 52eleqtrrd 2701 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
5453expr 642 . . . . . . 7 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂𝑏 ∈ ran 𝑂))
5554pm2.18d 124 . . . . . 6 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
5655ex 450 . . . . 5 (𝜑 → (𝑏𝐴𝑏 ∈ ran 𝑂))
5756ssrdv 3589 . . . 4 (𝜑𝐴 ⊆ ran 𝑂)
5811, 57eqssd 3600 . . 3 (𝜑 → ran 𝑂 = 𝐴)
59 isoeq5 6525 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
6058, 59syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
618, 60mpbid 222 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3554  wss 3555   class class class wbr 4613  cmpt 4673   E cep 4983   Se wse 5031   We wwe 5032  dom cdm 5074  ran crn 5075  cima 5077  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848  crio 6564  recscrecs 7412  OrdIsocoi 8358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-wrecs 7352  df-recs 7413  df-oi 8359
This theorem is referenced by:  ordtype  8381
  Copyright terms: Public domain W3C validator