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Theorem r1val3 9815
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 9744 . . . . 5 𝑅1 Fn On
21fndmi 6642 . . . 4 dom 𝑅1 = On
32eleq2i 2824 . . 3 (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4 r1val1 9763 . . 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 6378 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ On)
7 r1val2 9814 . . . . 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 4613 . . 3 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ 𝒫 (𝑅1β€˜π‘₯) = 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
109iuneq2dv 5014 . 2 (𝐴 ∈ On β†’ βˆͺ π‘₯ ∈ 𝐴 𝒫 (𝑅1β€˜π‘₯) = βˆͺ π‘₯ ∈ 𝐴 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
115, 10eqtrd 2771 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 2708  π’« cpw 4596  βˆͺ ciun 4990  dom cdm 5669  Oncon0 6353  β€˜cfv 6532  π‘…1cr1 9739  rankcrnk 9740
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-reg 9569  ax-inf2 9618
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-r1 9741  df-rank 9742
This theorem is referenced by: (None)
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