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Theorem ordtypelem10 8983
Description: Lemma for ordtype 8988. Using ax-rep 5186, 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 8981 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 8977 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6517 . . . 4 (𝜑 → ran 𝑂𝐴)
11 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏𝐴)
126adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
137adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
141, 2, 3, 4, 5, 12, 13ordtypelem8 8981 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
15 isof1o 7071 . . . . . . . . . . 11 (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
16 f1of 6611 . . . . . . . . . . 11 (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂⟶ran 𝑂)
1714, 15, 163syl 18 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂⟶ran 𝑂)
18 f1of1 6610 . . . . . . . . . . . 12 (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1→ran 𝑂)
1914, 15, 183syl 18 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂1-1→ran 𝑂)
20 simpl 483 . . . . . . . . . . . . 13 ((𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏𝐴)
21 seex 5516 . . . . . . . . . . . . 13 ((𝑅 Se 𝐴𝑏𝐴) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
227, 20, 21syl2an 595 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
2310adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂𝐴)
24 rexnal 3242 . . . . . . . . . . . . . . . . 17 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
251, 2, 3, 4, 5, 6, 7ordtypelem7 8980 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2625ord 860 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2726rexlimdva 3288 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2824, 27syl5bir 244 . . . . . . . . . . . . . . . 16 ((𝜑𝑏𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2928con1d 147 . . . . . . . . . . . . . . 15 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3029impr 455 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
319ffund 6514 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝑂)
3231funfnd 6382 . . . . . . . . . . . . . . . 16 (𝜑𝑂 Fn dom 𝑂)
3332adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
34 breq1 5065 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
3534ralrn 6849 . . . . . . . . . . . . . . 15 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3633, 35syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3730, 36mpbird 258 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
38 ssrab 4052 . . . . . . . . . . . . 13 (ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏} ↔ (ran 𝑂𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
3923, 37, 38sylanbrc 583 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏})
4022, 39ssexd 5224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
41 f1dmex 7652 . . . . . . . . . . 11 ((𝑂:dom 𝑂1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
4219, 40, 41syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
43 fex 6987 . . . . . . . . . 10 ((𝑂:dom 𝑂⟶ran 𝑂 ∧ dom 𝑂 ∈ V) → 𝑂 ∈ V)
4417, 42, 43syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
451, 2, 3, 4, 5, 12, 13, 44ordtypelem9 8982 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
46 isof1o 7071 . . . . . . . 8 (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂1-1-onto𝐴)
47 f1ofo 6618 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto𝐴𝑂:dom 𝑂onto𝐴)
48 forn 6589 . . . . . . . 8 (𝑂:dom 𝑂onto𝐴 → ran 𝑂 = 𝐴)
4945, 46, 47, 484syl 19 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
5011, 49eleqtrrd 2920 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
5150expr 457 . . . . 5 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂𝑏 ∈ ran 𝑂))
5251pm2.18d 127 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
5310, 52eqelssd 3991 . . 3 (𝜑 → ran 𝑂 = 𝐴)
54 isoeq5 7069 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
5553, 54syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
568, 55mpbid 233 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  wral 3142  wrex 3143  {crab 3146  Vcvv 3499  cin 3938  wss 3939   class class class wbr 5062  cmpt 5142   E cep 5462   Se wse 5510   We wwe 5511  dom cdm 5553  ran crn 5554  cima 5556  Oncon0 6188   Fn wfn 6346  wf 6347  1-1wf1 6348  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351   Isom wiso 6352  crio 7108  recscrecs 8001  OrdIsocoi 8965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-wrecs 7941  df-recs 8002  df-oi 8966
This theorem is referenced by:  ordtype  8988
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