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Theorem elmade2 33818
Description: Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
elmade2 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
Distinct variable groups:   𝐴,𝑙,𝑟   𝑋,𝑙,𝑟

Proof of Theorem elmade2
StepHypRef Expression
1 elmade 33817 . 2 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( M “ 𝐴)∃𝑟 ∈ 𝒫 ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
2 oldval 33804 . . . 4 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
32pweqd 4547 . . 3 (𝐴 ∈ On → 𝒫 ( O ‘𝐴) = 𝒫 ( M “ 𝐴))
43rexeqdv 3339 . . 3 (𝐴 ∈ On → (∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) ↔ ∃𝑟 ∈ 𝒫 ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
53, 4rexeqbidv 3327 . 2 (𝐴 ∈ On → (∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) ↔ ∃𝑙 ∈ 𝒫 ( M “ 𝐴)∃𝑟 ∈ 𝒫 ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
61, 5bitr4d 285 1 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  wrex 3063  𝒫 cpw 4528   cuni 4834   class class class wbr 5068  cima 5569  Oncon0 6231  cfv 6398  (class class class)co 7232   <<s csslt 33741   |s cscut 33743   M cmade 33792   O cold 33793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pow 5273  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-riota 7189  df-ov 7235  df-oprab 7236  df-mpo 7237  df-wrecs 8068  df-recs 8129  df-1o 8223  df-2o 8224  df-no 33612  df-slt 33613  df-bday 33614  df-sslt 33742  df-scut 33744  df-made 33797  df-old 33798
This theorem is referenced by:  madecut  33831  madebdayim  33836
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