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Theorem r1pw 9879
Description: A stronger property of 𝑅1 than rankpw 9877. The latter merely proves that 𝑅1 of the successor is a power set, but here we prove that if 𝐴 is in the cumulative hierarchy, then 𝒫 𝐴 is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1pw (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Proof of Theorem r1pw
StepHypRef Expression
1 rankpwi 9857 . . . . . 6 (𝐴 (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
21eleq1d 2811 . . . . 5 (𝐴 (𝑅1 “ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
3 eloni 6376 . . . . . . 7 (𝐵 ∈ On → Ord 𝐵)
4 ordsucelsuc 7821 . . . . . . 7 (Ord 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
53, 4syl 17 . . . . . 6 (𝐵 ∈ On → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ suc 𝐵))
65bicomd 222 . . . . 5 (𝐵 ∈ On → (suc (rank‘𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
72, 6sylan9bb 508 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → ((rank‘𝒫 𝐴) ∈ suc 𝐵 ↔ (rank‘𝐴) ∈ 𝐵))
8 pwwf 9841 . . . . . 6 (𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
98biimpi 215 . . . . 5 (𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
10 onsuc 7810 . . . . . 6 (𝐵 ∈ On → suc 𝐵 ∈ On)
11 r1fnon 9801 . . . . . . 7 𝑅1 Fn On
1211fndmi 6654 . . . . . 6 dom 𝑅1 = On
1310, 12eleqtrrdi 2837 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ dom 𝑅1)
14 rankr1ag 9836 . . . . 5 ((𝒫 𝐴 (𝑅1 “ On) ∧ suc 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
159, 13, 14syl2an 594 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝐴) ∈ suc 𝐵))
1612eleq2i 2818 . . . . 5 (𝐵 ∈ dom 𝑅1𝐵 ∈ On)
17 rankr1ag 9836 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
1816, 17sylan2br 593 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
197, 15, 183bitr4rd 311 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
2019ex 411 . 2 (𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
21 r1elwf 9830 . . . 4 (𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))
22 r1elwf 9830 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
23 r1elssi 9839 . . . . . 6 (𝒫 𝐴 (𝑅1 “ On) → 𝒫 𝐴 (𝑅1 “ On))
2422, 23syl 17 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 (𝑅1 “ On))
25 ssid 4002 . . . . . 6 𝐴𝐴
26 pwexr 7763 . . . . . . 7 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V)
27 elpwg 4601 . . . . . . 7 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
2826, 27syl 17 . . . . . 6 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → (𝐴 ∈ 𝒫 𝐴𝐴𝐴))
2925, 28mpbiri 257 . . . . 5 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ 𝒫 𝐴)
3024, 29sseldd 3980 . . . 4 (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 (𝑅1 “ On))
3121, 30pm5.21ni 376 . . 3 𝐴 (𝑅1 “ On) → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
3231a1d 25 . 2 𝐴 (𝑅1 “ On) → (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))
3320, 32pm2.61i 182 1 (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2099  Vcvv 3463  wss 3947  𝒫 cpw 4598   cuni 4906  dom cdm 5673  cima 5676  Ord word 6365  Oncon0 6366  suc csuc 6368  cfv 6544  𝑅1cr1 9796  rankcrnk 9797
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
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-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-r1 9798  df-rank 9799
This theorem is referenced by:  inatsk  10810
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