Theorem r1pwcl 9799

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 9748	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
2		elfvdm 6896	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
3	1, 2	jca 519	. . 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 9748	. . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
6		pwwf 9759	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
7	5, 6	sylibr 236	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
8		elfvdm 6896	. . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
9	7, 8	jca 519	. . 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 7824	. . . . . 6 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
12	11	adantr 484	. . . . 5 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13		rankpwi 9775	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
14	13	ad2antrl 738	. . . . . 6 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
15	14	eleq1d 2846	. . . . 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 9754	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
18	17	adantl 485	. . . 4 ⊢ ((Lim 𝐵 ∧ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1)) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
19		rankr1ag 9754	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
20	6, 19	sylanb 590	. . . . 5 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝒫 𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
21	20	adantl 485	. . . 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 416	. 2 ⊢ (Lim 𝐵 → ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))))
24	4, 10, 23	pm5.21ndd 381	1 ⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  𝒫 cpw 4552  ∪ cuni 4862  dom cdm 5643   “ cima 5646  Oncon0 6341  Lim wlim 6342  suc csuc 6343  ‘cfv 6516  𝑅₁cr1 9714  rankcrnk 9715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-om 7842  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-r1 9716  df-rank 9717

This theorem is referenced by:  r1limwun  10688