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Theorem aomclem5 36646
 Description: Lemma for dfac11 36650. 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 476 . . . . . 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 36645 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8 iftrue 4042 . . . . . . 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 2611 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
119, 10weeq12d 36628 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
127, 11mpbird 246 . . . 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 5650 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
19 orduniorsuc 6922 . . . . . . . 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 950 . . . . . 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 36644 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30 iffalse 4045 . . . . . . 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 2611 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
3331, 32weeq12d 36628 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
3429, 33mpbird 246 . . . 4 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
3512, 34pm2.61dan 828 . . 3 (𝜑 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36 weinxp 5109 . . 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 207 . 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 5026 . . 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 223 1 (𝜑𝐺 We (𝑅1‘dom 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∩ cint 4410   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   E cep 4947   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Ord word 5639  Oncon0 5640  suc csuc 5642  ‘cfv 5804  recscrecs 7354  Fincfn 7841  supcsup 8229  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-sup 8231  df-r1 8510  df-rank 8511 This theorem is referenced by:  aomclem6  36647
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