Theorem r1pwcl 9278

Description: The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
r1pwcl	⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))

Proof of Theorem r1pwcl

Step	Hyp	Ref	Expression
1		r1elwf 9227	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
2		elfvdm 6704	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
3	1, 2	jca 514	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
4	3	a1i 11	. 2 ⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
5		r1elwf 9227	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
6		pwwf 9238	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylibr 236	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
8		elfvdm 6704	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
9	7, 8	jca 514	. . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1))
10	9	a1i 11	. 2 ⊢ (Lim 𝐵 → (𝒫 𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)))
11		limsuc 7566	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
12	11	adantr 483	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13		rankpwi 9254	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
14	13	ad2antrl 726	. . . . . 6 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
15	14	eleq1d 2899	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
16	12, 15	bitr4d 284	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17		rankr1ag 9233	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	17	adantl 484	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19		rankr1ag 9233	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
20	6, 19	sylanb 583	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
21	20	adantl 484	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
22	16, 18, 21	3bitr4d 313	. . 3 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))
23	22	ex 415	. 2 ⊢ (Lim 𝐵 → ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))))
24	4, 10, 23	pm5.21ndd 383	1 ⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  𝒫 cpw 4541  ∪ cuni 4840  dom cdm 5557   “ cima 5560  Oncon0 6193  Lim wlim 6194  suc csuc 6195  ‘cfv 6357  𝑅₁cr1 9193  rankcrnk 9194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-r1 9195  df-rank 9196

This theorem is referenced by:  r1limwun  10160