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Theorem ordtypelem10 9517
Description: Lemma for ordtype 9522. Using ax-rep 5283, 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 9515 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 9511 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6721 . . . 4 (𝜑 → ran 𝑂𝐴)
11 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏𝐴)
126adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
137adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
149ffund 6717 . . . . . . . . . . . 12 (𝜑 → Fun 𝑂)
1514funfnd 6575 . . . . . . . . . . 11 (𝜑𝑂 Fn dom 𝑂)
1615adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
171, 2, 3, 4, 5, 12, 13ordtypelem8 9515 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
18 isof1o 7314 . . . . . . . . . . . 12 (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
19 f1of1 6828 . . . . . . . . . . . 12 (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1→ran 𝑂)
2017, 18, 193syl 18 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂1-1→ran 𝑂)
21 simpl 484 . . . . . . . . . . . . 13 ((𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏𝐴)
22 seex 5636 . . . . . . . . . . . . 13 ((𝑅 Se 𝐴𝑏𝐴) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
237, 21, 22syl2an 597 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
2410adantr 482 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂𝐴)
25 rexnal 3101 . . . . . . . . . . . . . . . . 17 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
261, 2, 3, 4, 5, 6, 7ordtypelem7 9514 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2726ord 863 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2827rexlimdva 3156 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2925, 28biimtrrid 242 . . . . . . . . . . . . . . . 16 ((𝜑𝑏𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
3029con1d 145 . . . . . . . . . . . . . . 15 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3130impr 456 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
32 breq1 5149 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
3332ralrn 7084 . . . . . . . . . . . . . . 15 (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3416, 33syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3531, 34mpbird 257 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)
36 ssrab 4068 . . . . . . . . . . . . 13 (ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏} ↔ (ran 𝑂𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
3724, 35, 36sylanbrc 584 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏})
3823, 37ssexd 5322 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
39 f1dmex 7937 . . . . . . . . . . 11 ((𝑂:dom 𝑂1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
4020, 38, 39syl2anc 585 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
4116, 40fnexd 7214 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
421, 2, 3, 4, 5, 12, 13, 41ordtypelem9 9516 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
43 isof1o 7314 . . . . . . . 8 (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂1-1-onto𝐴)
44 f1ofo 6836 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto𝐴𝑂:dom 𝑂onto𝐴)
45 forn 6804 . . . . . . . 8 (𝑂:dom 𝑂onto𝐴 → ran 𝑂 = 𝐴)
4642, 43, 44, 454syl 19 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
4711, 46eleqtrrd 2837 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
4847expr 458 . . . . 5 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂𝑏 ∈ ran 𝑂))
4948pm2.18d 127 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
5010, 49eqelssd 4001 . . 3 (𝜑 → ran 𝑂 = 𝐴)
51 isoeq5 7312 . . 3 (ran 𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
5250, 51syl 17 . 2 (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)))
538, 52mpbid 231 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cin 3945  wss 3946   class class class wbr 5146  cmpt 5229   E cep 5577   Se wse 5627   We wwe 5628  dom cdm 5674  ran crn 5675  cima 5677  Oncon0 6360   Fn wfn 6534  1-1wf1 6536  ontowfo 6537  1-1-ontowf1o 6538  cfv 6539   Isom wiso 6540  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-rep 5283  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-isom 6548  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:  ordtype  9522
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