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Theorem r1val3 9817
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 β†’ (𝑅1β€˜π΄) = βˆͺ π‘₯ ∈ 𝐴 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
Distinct variable group:   π‘₯,𝑦,𝐴

Proof of Theorem r1val3
StepHypRef Expression
1 r1fnon 9746 . . . . 5 𝑅1 Fn On
21fndmi 6643 . . . 4 dom 𝑅1 = On
32eleq2i 2825 . . 3 (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4 r1val1 9765 . . 3 (𝐴 ∈ dom 𝑅1 β†’ (𝑅1β€˜π΄) = βˆͺ π‘₯ ∈ 𝐴 𝒫 (𝑅1β€˜π‘₯))
53, 4sylbir 234 . 2 (𝐴 ∈ On β†’ (𝑅1β€˜π΄) = βˆͺ π‘₯ ∈ 𝐴 𝒫 (𝑅1β€˜π‘₯))
6 onelon 6379 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ On)
7 r1val2 9816 . . . . 5 (π‘₯ ∈ On β†’ (𝑅1β€˜π‘₯) = {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
86, 7syl 17 . . . 4 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ (𝑅1β€˜π‘₯) = {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
98pweqd 4614 . . 3 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ 𝒫 (𝑅1β€˜π‘₯) = 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
109iuneq2dv 5015 . 2 (𝐴 ∈ On β†’ βˆͺ π‘₯ ∈ 𝐴 𝒫 (𝑅1β€˜π‘₯) = βˆͺ π‘₯ ∈ 𝐴 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
115, 10eqtrd 2772 1 (𝐴 ∈ On β†’ (𝑅1β€˜π΄) = βˆͺ π‘₯ ∈ 𝐴 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  π’« cpw 4597  βˆͺ ciun 4991  dom cdm 5670  Oncon0 6354  β€˜cfv 6533  π‘…1cr1 9741  rankcrnk 9742
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-reg 9571  ax-inf2 9620
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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-r1 9743  df-rank 9744
This theorem is referenced by: (None)
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