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Theorem r1val3 8561
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 8490 . . . . 5 𝑅1 Fn On
2 fndm 5890 . . . . 5 (𝑅1 Fn On → dom 𝑅1 = On)
31, 2ax-mp 5 . . . 4 dom 𝑅1 = On
43eleq2i 2679 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
5 r1val1 8509 . . 3 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
64, 5sylbir 223 . 2 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
7 onelon 5651 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
8 r1val2 8560 . . . . 5 (𝑥 ∈ On → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
97, 8syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
109pweqd 4112 . . 3 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝒫 (𝑅1𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
1110iuneq2dv 4472 . 2 (𝐴 ∈ On → 𝑥𝐴 𝒫 (𝑅1𝑥) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
126, 11eqtrd 2643 1 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {cab 2595  𝒫 cpw 4107   ciun 4449  dom cdm 5028  Oncon0 5626   Fn wfn 5785  cfv 5790  𝑅1cr1 8485  rankcrnk 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-r1 8487  df-rank 8488
This theorem is referenced by: (None)
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