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Theorem ordtypelem10 9564
Description: Lemma for ordtype 9569. Using ax-rep 5284, 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 9562 . 2 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
91, 2, 3, 4, 5, 6, 7ordtypelem4 9558 . . . . 5 (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
109frnd 6744 . . . 4 (𝜑 → ran 𝑂𝐴)
11 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏𝐴)
126adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴)
137adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴)
149ffund 6740 . . . . . . . . . . . 12 (𝜑 → Fun 𝑂)
1514funfnd 6598 . . . . . . . . . . 11 (𝜑𝑂 Fn dom 𝑂)
1615adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂)
171, 2, 3, 4, 5, 12, 13ordtypelem8 9562 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
18 isof1o 7342 . . . . . . . . . . . 12 (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
19 f1of1 6847 . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 ((𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏𝐴)
22 seex 5647 . . . . . . . . . . . . 13 ((𝑅 Se 𝐴𝑏𝐴) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
237, 21, 22syl2an 596 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐𝐴𝑐𝑅𝑏} ∈ V)
2410adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂𝐴)
25 rexnal 3097 . . . . . . . . . . . . . . . . 17 (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
261, 2, 3, 4, 5, 6, 7ordtypelem7 9561 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2726ord 864 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑏𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2827rexlimdva 3152 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
2925, 28biimtrrid 243 . . . . . . . . . . . . . . . 16 ((𝜑𝑏𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏𝑏 ∈ ran 𝑂))
3029con1d 145 . . . . . . . . . . . . . . 15 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏))
3130impr 454 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂𝑚)𝑅𝑏)
32 breq1 5150 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑂𝑚) → (𝑐𝑅𝑏 ↔ (𝑂𝑚)𝑅𝑏))
3332ralrn 7107 . . . . . . . . . . . . . . 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 4082 . . . . . . . . . . . . 13 (ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏} ↔ (ran 𝑂𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏))
3724, 35, 36sylanbrc 583 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐𝐴𝑐𝑅𝑏})
3823, 37ssexd 5329 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V)
39 f1dmex 7979 . . . . . . . . . . 11 ((𝑂:dom 𝑂1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V)
4020, 38, 39syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V)
4116, 40fnexd 7237 . . . . . . . . 9 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V)
421, 2, 3, 4, 5, 12, 13, 41ordtypelem9 9563 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
43 isof1o 7342 . . . . . . . 8 (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂1-1-onto𝐴)
44 f1ofo 6855 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto𝐴𝑂:dom 𝑂onto𝐴)
45 forn 6823 . . . . . . . 8 (𝑂:dom 𝑂onto𝐴 → ran 𝑂 = 𝐴)
4642, 43, 44, 454syl 19 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴)
4711, 46eleqtrrd 2841 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂)
4847expr 456 . . . . 5 ((𝜑𝑏𝐴) → (¬ 𝑏 ∈ ran 𝑂𝑏 ∈ ran 𝑂))
4948pm2.18d 127 . . . 4 ((𝜑𝑏𝐴) → 𝑏 ∈ ran 𝑂)
5010, 49eqelssd 4016 . . 3 (𝜑 → ran 𝑂 = 𝐴)
51 isoeq5 7340 . . 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 232 1 (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cin 3961  wss 3962   class class class wbr 5147  cmpt 5230   E cep 5587   Se wse 5638   We wwe 5639  dom cdm 5688  ran crn 5689  cima 5691  Oncon0 6385   Fn wfn 6557  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563  crio 7386  recscrecs 8408  OrdIsocoi 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-oi 9547
This theorem is referenced by:  ordtype  9569
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