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Theorem aomclem5 37143
Description: Lemma for dfac11 37147. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem5.b 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
aomclem5.c 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
aomclem5.d 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
aomclem5.e 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
aomclem5.f 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
aomclem5.g 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
aomclem5.on (𝜑 → dom 𝑧 ∈ On)
aomclem5.we (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
aomclem5.a (𝜑𝐴 ∈ On)
aomclem5.za (𝜑 → dom 𝑧𝐴)
aomclem5.y (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
Assertion
Ref Expression
aomclem5 (𝜑𝐺 We (𝑅1‘dom 𝑧))
Distinct variable groups:   𝑦,𝑧,𝑎,𝑏,𝑐,𝑑   𝜑,𝑎,𝑏   𝐶,𝑎,𝑏,𝑐,𝑑   𝐷,𝑎,𝑏,𝑐,𝑑
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑐,𝑑)   𝐴(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aomclem5
StepHypRef Expression
1 aomclem5.f . . . . . 6 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}
2 aomclem5.on . . . . . . 7 (𝜑 → dom 𝑧 ∈ On)
32adantr 481 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
4 simpr 477 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 = dom 𝑧)
5 aomclem5.we . . . . . . 7 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
65adantr 481 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
71, 3, 4, 6aomclem4 37142 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8 iftrue 4069 . . . . . . 7 (dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
98adantl 482 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
10 eqidd 2622 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
119, 10weeq12d 37125 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
127, 11mpbird 247 . . . 4 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
13 aomclem5.b . . . . . 6 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
14 aomclem5.c . . . . . 6 𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))
15 aomclem5.d . . . . . 6 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))
16 aomclem5.e . . . . . 6 𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}
172adantr 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
18 eloni 5697 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
19 orduniorsuc 6984 . . . . . . . 8 (Ord dom 𝑧 → (dom 𝑧 = dom 𝑧 ∨ dom 𝑧 = suc dom 𝑧))
202, 18, 193syl 18 . . . . . . 7 (𝜑 → (dom 𝑧 = dom 𝑧 ∨ dom 𝑧 = suc dom 𝑧))
2120orcanai 951 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 = suc dom 𝑧)
225adantr 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
23 aomclem5.a . . . . . . 7 (𝜑𝐴 ∈ On)
2423adantr 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐴 ∈ On)
25 aomclem5.za . . . . . . 7 (𝜑 → dom 𝑧𝐴)
2625adantr 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧𝐴)
27 aomclem5.y . . . . . . 7 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2827adantr 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 37141 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30 iffalse 4072 . . . . . . 7 (¬ dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
3130adantl 482 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
32 eqidd 2622 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
3331, 32weeq12d 37125 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
3429, 33mpbird 247 . . . 4 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
3512, 34pm2.61dan 831 . . 3 (𝜑 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36 weinxp 5152 . . 3 (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
3735, 36sylib 208 . 2 (𝜑 → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
38 aomclem5.g . . 3 𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))
39 weeq1 5067 . . 3 (𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) → (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧)))
4038, 39ax-mp 5 . 2 (𝐺 We (𝑅1‘dom 𝑧) ↔ (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) We (𝑅1‘dom 𝑧))
4137, 40sylibr 224 1 (𝜑𝐺 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  cin 3558  wss 3559  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153   cuni 4407   cint 4445   class class class wbr 4618  {copab 4677  cmpt 4678   E cep 4988   We wwe 5037   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Ord word 5686  Oncon0 5687  suc csuc 5689  cfv 5852  recscrecs 7419  Fincfn 7907  supcsup 8298  𝑅1cr1 8577  rankcrnk 8578
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-er 7694  df-map 7811  df-en 7908  df-fin 7911  df-sup 8300  df-r1 8579  df-rank 8580
This theorem is referenced by:  aomclem6  37144
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