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Theorem r1val3 8584
 Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
r1val3 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem r1val3
StepHypRef Expression
1 r1fnon 8513 . . . . 5 𝑅1 Fn On
2 fndm 5904 . . . . 5 (𝑅1 Fn On → dom 𝑅1 = On)
31, 2ax-mp 5 . . . 4 dom 𝑅1 = On
43eleq2i 2680 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
5 r1val1 8532 . . 3 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
64, 5sylbir 224 . 2 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
7 onelon 5665 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
8 r1val2 8583 . . . . 5 (𝑥 ∈ On → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
97, 8syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
109pweqd 4113 . . 3 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝒫 (𝑅1𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
1110iuneq2dv 4478 . 2 (𝐴 ∈ On → 𝑥𝐴 𝒫 (𝑅1𝑥) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
126, 11eqtrd 2644 1 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  𝒫 cpw 4108  ∪ ciun 4455  dom cdm 5038  Oncon0 5640   Fn wfn 5799  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by: (None)
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