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Theorem aomclem5 42719
Description: Lemma for dfac11 42723. 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 479 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
4 simpr 483 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 = dom 𝑧)
5 aomclem5.we . . . . . . 7 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
65adantr 479 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
71, 3, 4, 6aomclem4 42718 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8 iftrue 4539 . . . . . . 7 (dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
98adantl 480 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
10 eqidd 2727 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
119, 10weeq12d 42701 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
127, 11mpbird 256 . . . 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 479 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
18 eloni 6386 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
19 orduniorsuc 7839 . . . . . . . 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 1000 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 = suc dom 𝑧)
225adantr 479 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
23 aomclem5.a . . . . . . 7 (𝜑𝐴 ∈ On)
2423adantr 479 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐴 ∈ On)
25 aomclem5.za . . . . . . 7 (𝜑 → dom 𝑧𝐴)
2625adantr 479 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧𝐴)
27 aomclem5.y . . . . . . 7 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2827adantr 479 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 42717 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30 iffalse 4542 . . . . . . 7 (¬ dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
3130adantl 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
32 eqidd 2727 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
3331, 32weeq12d 42701 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
3429, 33mpbird 256 . . . 4 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
3512, 34pm2.61dan 811 . . 3 (𝜑 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36 weinxp 5766 . . 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 217 . 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 5670 . . 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 233 1 (𝜑𝐺 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  cdif 3944  cin 3946  wss 3947  c0 4325  ifcif 4533  𝒫 cpw 4607  {csn 4633   cuni 4913   cint 4954   class class class wbr 5153  {copab 5215  cmpt 5236   E cep 5585   We wwe 5636   × cxp 5680  ccnv 5681  dom cdm 5682  ran crn 5683  cima 5685  Ord word 6375  Oncon0 6376  suc csuc 6378  cfv 6554  recscrecs 8400  Fincfn 8974  supcsup 9483  𝑅1cr1 9805  rankcrnk 9806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-map 8857  df-en 8975  df-fin 8978  df-sup 9485  df-r1 9807  df-rank 9808
This theorem is referenced by:  aomclem6  42720
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