Theorem r1pwcl 9866

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 9815	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
2		elfvdm 6918	. . . 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 9815	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
6		pwwf 9826	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylibr 234	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
8		elfvdm 6918	. . . 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 7849	. . . . . 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 9842	. . . . . . 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 2820	. . . . 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 9821	. . . . 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 9821	. . . . . 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 1540   ∈ wcel 2109  𝒫 cpw 4580  ∪ cuni 4888  dom cdm 5659   “ cima 5662  Oncon0 6357  Lim wlim 6358  suc csuc 6359  ‘cfv 6536  𝑅₁cr1 9781  rankcrnk 9782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9783  df-rank 9784

This theorem is referenced by:  r1limwun  10755