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Theorem r1val3 8877
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 8806 . . . . 5 𝑅1 Fn On
2 fndm 6152 . . . . 5 (𝑅1 Fn On → dom 𝑅1 = On)
31, 2ax-mp 5 . . . 4 dom 𝑅1 = On
43eleq2i 2832 . . 3 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
5 r1val1 8825 . . 3 (𝐴 ∈ dom 𝑅1 → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
64, 5sylbir 225 . 2 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))
7 onelon 5910 . . . . 5 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
8 r1val2 8876 . . . . 5 (𝑥 ∈ On → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
97, 8syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝑥𝐴) → (𝑅1𝑥) = {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
109pweqd 4308 . . 3 ((𝐴 ∈ On ∧ 𝑥𝐴) → 𝒫 (𝑅1𝑥) = 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
1110iuneq2dv 4695 . 2 (𝐴 ∈ On → 𝑥𝐴 𝒫 (𝑅1𝑥) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
126, 11eqtrd 2795 1 (𝐴 ∈ On → (𝑅1𝐴) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  {cab 2747  𝒫 cpw 4303   ciun 4673  dom cdm 5267  Oncon0 5885   Fn wfn 6045  cfv 6050  𝑅1cr1 8801  rankcrnk 8802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-reg 8665  ax-inf2 8714
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-om 7233  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-r1 8803  df-rank 8804
This theorem is referenced by: (None)
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