Theorem r1pwcl 9844

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 9793	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
2		elfvdm 6928	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
3	1, 2	jca 512	. . 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 9793	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
6		pwwf 9804	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylibr 233	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
8		elfvdm 6928	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
9	7, 8	jca 512	. . 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 7840	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
12	11	adantr 481	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13		rankpwi 9820	. . . . . . 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 2818	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
16	12, 15	bitr4d 281	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
17		rankr1ag 9799	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	17	adantl 482	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19		rankr1ag 9799	. . . . . 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 482	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
22	16, 18, 21	3bitr4d 310	. . 3 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))
23	22	ex 413	. 2 ⊢ (Lim 𝐵 → ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))))
24	4, 10, 23	pm5.21ndd 380	1 ⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  𝒫 cpw 4602  ∪ cuni 4908  dom cdm 5676   “ cima 5679  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ‘cfv 6543  𝑅₁cr1 9759  rankcrnk 9760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762

This theorem is referenced by:  r1limwun  10733