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Theorem aomclem5 42531
Description: Lemma for dfac11 42535. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem5.b 𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}
aomclem5.c 𝐢 = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))
aomclem5.d 𝐷 = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))
aomclem5.e 𝐸 = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}
aomclem5.f 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
aomclem5.g 𝐺 = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))
aomclem5.on (πœ‘ β†’ dom 𝑧 ∈ On)
aomclem5.we (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
aomclem5.a (πœ‘ β†’ 𝐴 ∈ On)
aomclem5.za (πœ‘ β†’ dom 𝑧 βŠ† 𝐴)
aomclem5.y (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
Assertion
Ref Expression
aomclem5 (πœ‘ β†’ 𝐺 We (𝑅1β€˜dom 𝑧))
Distinct variable groups:   𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑   πœ‘,π‘Ž,𝑏   𝐢,π‘Ž,𝑏,𝑐,𝑑   𝐷,π‘Ž,𝑏,𝑐,𝑑
Allowed substitution hints:   πœ‘(𝑦,𝑧,𝑐,𝑑)   𝐴(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐡(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐢(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)

Proof of Theorem aomclem5
StepHypRef Expression
1 aomclem5.f . . . . . 6 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
2 aomclem5.on . . . . . . 7 (πœ‘ β†’ dom 𝑧 ∈ On)
32adantr 479 . . . . . 6 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ dom 𝑧 ∈ On)
4 simpr 483 . . . . . 6 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ dom 𝑧 = βˆͺ dom 𝑧)
5 aomclem5.we . . . . . . 7 (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
65adantr 479 . . . . . 6 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
71, 3, 4, 6aomclem4 42530 . . . . 5 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ 𝐹 We (𝑅1β€˜dom 𝑧))
8 iftrue 4538 . . . . . . 7 (dom 𝑧 = βˆͺ dom 𝑧 β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) = 𝐹)
98adantl 480 . . . . . 6 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) = 𝐹)
10 eqidd 2729 . . . . . 6 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ (𝑅1β€˜dom 𝑧) = (𝑅1β€˜dom 𝑧))
119, 10weeq12d 42513 . . . . 5 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧) ↔ 𝐹 We (𝑅1β€˜dom 𝑧)))
127, 11mpbird 256 . . . 4 ((πœ‘ ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧))
13 aomclem5.b . . . . . 6 𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}
14 aomclem5.c . . . . . 6 𝐢 = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))
15 aomclem5.d . . . . . 6 𝐷 = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))
16 aomclem5.e . . . . . 6 𝐸 = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}
172adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ dom 𝑧 ∈ On)
18 eloni 6384 . . . . . . . 8 (dom 𝑧 ∈ On β†’ Ord dom 𝑧)
19 orduniorsuc 7841 . . . . . . . 8 (Ord dom 𝑧 β†’ (dom 𝑧 = βˆͺ dom 𝑧 ∨ dom 𝑧 = suc βˆͺ dom 𝑧))
202, 18, 193syl 18 . . . . . . 7 (πœ‘ β†’ (dom 𝑧 = βˆͺ dom 𝑧 ∨ dom 𝑧 = suc βˆͺ dom 𝑧))
2120orcanai 1000 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ dom 𝑧 = suc βˆͺ dom 𝑧)
225adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
23 aomclem5.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ On)
2423adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ 𝐴 ∈ On)
25 aomclem5.za . . . . . . 7 (πœ‘ β†’ dom 𝑧 βŠ† 𝐴)
2625adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ dom 𝑧 βŠ† 𝐴)
27 aomclem5.y . . . . . . 7 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
2827adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
2913, 14, 15, 16, 17, 21, 22, 24, 26, 28aomclem3 42529 . . . . 5 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ 𝐸 We (𝑅1β€˜dom 𝑧))
30 iffalse 4541 . . . . . . 7 (Β¬ dom 𝑧 = βˆͺ dom 𝑧 β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) = 𝐸)
3130adantl 480 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) = 𝐸)
32 eqidd 2729 . . . . . 6 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ (𝑅1β€˜dom 𝑧) = (𝑅1β€˜dom 𝑧))
3331, 32weeq12d 42513 . . . . 5 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧) ↔ 𝐸 We (𝑅1β€˜dom 𝑧)))
3429, 33mpbird 256 . . . 4 ((πœ‘ ∧ Β¬ dom 𝑧 = βˆͺ dom 𝑧) β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧))
3512, 34pm2.61dan 811 . . 3 (πœ‘ β†’ if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧))
36 weinxp 5766 . . 3 (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) We (𝑅1β€˜dom 𝑧) ↔ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) We (𝑅1β€˜dom 𝑧))
3735, 36sylib 217 . 2 (πœ‘ β†’ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) We (𝑅1β€˜dom 𝑧))
38 aomclem5.g . . 3 𝐺 = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))
39 weeq1 5670 . . 3 (𝐺 = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) β†’ (𝐺 We (𝑅1β€˜dom 𝑧) ↔ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) We (𝑅1β€˜dom 𝑧)))
4038, 39ax-mp 5 . 2 (𝐺 We (𝑅1β€˜dom 𝑧) ↔ (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) We (𝑅1β€˜dom 𝑧))
4137, 40sylibr 233 1 (πœ‘ β†’ 𝐺 We (𝑅1β€˜dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4326  ifcif 4532  π’« cpw 4606  {csn 4632  βˆͺ cuni 4912  βˆ© cint 4953   class class class wbr 5152  {copab 5214   ↦ cmpt 5235   E cep 5585   We wwe 5636   Γ— cxp 5680  β—‘ccnv 5681  dom cdm 5682  ran crn 5683   β€œ cima 5685  Ord word 6373  Oncon0 6374  suc csuc 6376  β€˜cfv 6553  recscrecs 8399  Fincfn 8972  supcsup 9473  π‘…1cr1 9795  rankcrnk 9796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  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 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-map 8855  df-en 8973  df-fin 8976  df-sup 9475  df-r1 9797  df-rank 9798
This theorem is referenced by:  aomclem6  42532
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