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Theorem r1val3 9814
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 9743 . . . . 5 𝑅1 Fn On
21fndmi 6641 . . . 4 dom 𝑅1 = On
32eleq2i 2824 . . 3 (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ On)
4 r1val1 9762 . . 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 6377 . . . . 5 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ On)
7 r1val2 9813 . . . . 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 4612 . . 3 ((𝐴 ∈ On ∧ π‘₯ ∈ 𝐴) β†’ 𝒫 (𝑅1β€˜π‘₯) = 𝒫 {𝑦 ∣ (rankβ€˜π‘¦) ∈ π‘₯})
109iuneq2dv 5013 . 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 4595  βˆͺ ciun 4989  dom cdm 5668  Oncon0 6352  β€˜cfv 6531  π‘…1cr1 9738  rankcrnk 9739
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 5277  ax-sep 5291  ax-nul 5298  ax-pow 5355  ax-pr 5419  ax-un 7707  ax-reg 9568  ax-inf2 9617
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 3474  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4943  df-iun 4991  df-br 5141  df-opab 5203  df-mpt 5224  df-tr 5258  df-id 5566  df-eprel 5572  df-po 5580  df-so 5581  df-fr 5623  df-we 5625  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-pred 6288  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7395  df-om 7838  df-2nd 7957  df-frecs 8247  df-wrecs 8278  df-recs 8352  df-rdg 8391  df-r1 9740  df-rank 9741
This theorem is referenced by: (None)
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