Theorem r1pwcl 9888

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 9837	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
2		elfvdm 6942	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
3	1, 2	jca 511	. . 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 9837	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
6		pwwf 9848	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylibr 234	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
8		elfvdm 6942	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
9	7, 8	jca 511	. . 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 7871	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
12	11	adantr 480	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13		rankpwi 9864	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
14	13	ad2antrl 728	. . . . . 6 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
15	14	eleq1d 2825	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
16	12, 15	bitr4d 282	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17		rankr1ag 9843	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	17	adantl 481	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19		rankr1ag 9843	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
20	6, 19	sylanb 581	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
21	20	adantl 481	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
22	16, 18, 21	3bitr4d 311	. . 3 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))
23	22	ex 412	. 2 ⊢ (Lim 𝐵 → ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))))
24	4, 10, 23	pm5.21ndd 379	1 ⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  𝒫 cpw 4599  ∪ cuni 4906  dom cdm 5684   “ cima 5687  Oncon0 6383  Lim wlim 6384  suc csuc 6385  ‘cfv 6560  𝑅₁cr1 9803  rankcrnk 9804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806

This theorem is referenced by:  r1limwun  10777