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Theorem aomclem5 38471
Description: Lemma for dfac11 38475. 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 474 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
4 simpr 479 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → dom 𝑧 = dom 𝑧)
5 aomclem5.we . . . . . . 7 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
65adantr 474 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
71, 3, 4, 6aomclem4 38470 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → 𝐹 We (𝑅1‘dom 𝑧))
8 iftrue 4312 . . . . . . 7 (dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
98adantl 475 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐹)
10 eqidd 2826 . . . . . 6 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
119, 10weeq12d 38453 . . . . 5 ((𝜑 ∧ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐹 We (𝑅1‘dom 𝑧)))
127, 11mpbird 249 . . . 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 474 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 ∈ On)
18 eloni 5973 . . . . . . . 8 (dom 𝑧 ∈ On → Ord dom 𝑧)
19 orduniorsuc 7291 . . . . . . . 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 1032 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧 = suc dom 𝑧)
225adantr 474 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
23 aomclem5.a . . . . . . 7 (𝜑𝐴 ∈ On)
2423adantr 474 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐴 ∈ On)
25 aomclem5.za . . . . . . 7 (𝜑 → dom 𝑧𝐴)
2625adantr 474 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → dom 𝑧𝐴)
27 aomclem5.y . . . . . . 7 (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2827adantr 474 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 38469 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → 𝐸 We (𝑅1‘dom 𝑧))
30 iffalse 4315 . . . . . . 7 (¬ dom 𝑧 = dom 𝑧 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
3130adantl 475 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) = 𝐸)
32 eqidd 2826 . . . . . 6 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (𝑅1‘dom 𝑧) = (𝑅1‘dom 𝑧))
3331, 32weeq12d 38453 . . . . 5 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧) ↔ 𝐸 We (𝑅1‘dom 𝑧)))
3429, 33mpbird 249 . . . 4 ((𝜑 ∧ ¬ dom 𝑧 = dom 𝑧) → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
3512, 34pm2.61dan 849 . . 3 (𝜑 → if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) We (𝑅1‘dom 𝑧))
36 weinxp 5421 . . 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 210 . 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 5330 . . 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 226 1 (𝜑𝐺 We (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 880   = wceq 1658  wcel 2166  wne 2999  wral 3117  wrex 3118  Vcvv 3414  cdif 3795  cin 3797  wss 3798  c0 4144  ifcif 4306  𝒫 cpw 4378  {csn 4397   cuni 4658   cint 4697   class class class wbr 4873  {copab 4935  cmpt 4952   E cep 5254   We wwe 5300   × cxp 5340  ccnv 5341  dom cdm 5342  ran crn 5343  cima 5345  Ord word 5962  Oncon0 5963  suc csuc 5965  cfv 6123  recscrecs 7733  Fincfn 8222  supcsup 8615  𝑅1cr1 8902  rankcrnk 8903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-er 8009  df-map 8124  df-en 8223  df-fin 8226  df-sup 8617  df-r1 8904  df-rank 8905
This theorem is referenced by:  aomclem6  38472
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