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Theorem aomclem5 43047
Description: Lemma for dfac11 43051. 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 480 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
4 simpr 484 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 = dom 𝑧)
5 aomclem5.we . . . . . . 7 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
65adantr 480 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
71, 3, 4, 6aomclem4 43046 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8 iftrue 4537 . . . . . . 7 (dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
98adantl 481 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
10 eqidd 2736 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
119, 10weeq12d 5678 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
127, 11mpbird 257 . . . 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 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
18 eloni 6396 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
19 orduniorsuc 7850 . . . . . . . 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 1004 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 = suc dom 𝑧)
225adantr 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
23 aomclem5.a . . . . . . 7 (𝜑𝐴 ∈ On)
2423adantr 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐴 ∈ On)
25 aomclem5.za . . . . . . 7 (𝜑 → dom 𝑧𝐴)
2625adantr 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧𝐴)
27 aomclem5.y . . . . . . 7 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2827adantr 480 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 43045 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30 iffalse 4540 . . . . . . 7 (¬ dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
3130adantl 481 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
32 eqidd 2736 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
3331, 32weeq12d 5678 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
3429, 33mpbird 257 . . . 4 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
3512, 34pm2.61dan 813 . . 3 (𝜑 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36 weinxp 5773 . . 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 218 . 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 5676 . . 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 234 1 (𝜑𝐺 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cin 3962  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631   cuni 4912   cint 4951   class class class wbr 5148  {copab 5210  cmpt 5231   E cep 5588   We wwe 5640   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Ord word 6385  Oncon0 6386  suc csuc 6388  cfv 6563  recscrecs 8409  Fincfn 8984  supcsup 9478  𝑅1cr1 9800  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-en 8985  df-fin 8988  df-sup 9480  df-r1 9802  df-rank 9803
This theorem is referenced by:  aomclem6  43048
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