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Theorem elold 27908
Description: Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
elold (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
Distinct variable groups:   𝐴,𝑏   𝑋,𝑏

Proof of Theorem elold
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oldval 27893 . . 3 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
21eleq2d 2827 . 2 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ 𝑋 ( M “ 𝐴)))
3 eluni 4910 . . 3 (𝑋 ( M “ 𝐴) ↔ ∃𝑦(𝑋𝑦𝑦 ∈ ( M “ 𝐴)))
4 madef 27895 . . . . . . . 8 M :On⟶𝒫 No
5 ffn 6736 . . . . . . . 8 ( M :On⟶𝒫 No → M Fn On)
64, 5ax-mp 5 . . . . . . 7 M Fn On
7 onss 7805 . . . . . . 7 (𝐴 ∈ On → 𝐴 ⊆ On)
8 fvelimab 6981 . . . . . . 7 (( M Fn On ∧ 𝐴 ⊆ On) → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦))
96, 7, 8sylancr 587 . . . . . 6 (𝐴 ∈ On → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦))
109anbi2d 630 . . . . 5 (𝐴 ∈ On → ((𝑋𝑦𝑦 ∈ ( M “ 𝐴)) ↔ (𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦)))
1110exbidv 1921 . . . 4 (𝐴 ∈ On → (∃𝑦(𝑋𝑦𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑦(𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦)))
12 fvex 6919 . . . . . . 7 ( M ‘𝑏) ∈ V
1312clel3 3662 . . . . . 6 (𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦))
1413rexbii 3094 . . . . 5 (∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑏𝐴𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦))
15 rexcom4 3288 . . . . 5 (∃𝑏𝐴𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦) ↔ ∃𝑦𝑏𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦))
16 eqcom 2744 . . . . . . . . 9 (𝑦 = ( M ‘𝑏) ↔ ( M ‘𝑏) = 𝑦)
1716anbi2ci 625 . . . . . . . 8 ((𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦) ↔ (𝑋𝑦 ∧ ( M ‘𝑏) = 𝑦))
1817rexbii 3094 . . . . . . 7 (∃𝑏𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦) ↔ ∃𝑏𝐴 (𝑋𝑦 ∧ ( M ‘𝑏) = 𝑦))
19 r19.42v 3191 . . . . . . 7 (∃𝑏𝐴 (𝑋𝑦 ∧ ( M ‘𝑏) = 𝑦) ↔ (𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦))
2018, 19bitri 275 . . . . . 6 (∃𝑏𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦) ↔ (𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦))
2120exbii 1848 . . . . 5 (∃𝑦𝑏𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋𝑦) ↔ ∃𝑦(𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦))
2214, 15, 213bitrri 298 . . . 4 (∃𝑦(𝑋𝑦 ∧ ∃𝑏𝐴 ( M ‘𝑏) = 𝑦) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏))
2311, 22bitrdi 287 . . 3 (𝐴 ∈ On → (∃𝑦(𝑋𝑦𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
243, 23bitrid 283 . 2 (𝐴 ∈ On → (𝑋 ( M “ 𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
252, 24bitrd 279 1 (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏𝐴 𝑋 ∈ ( M ‘𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  wss 3951  𝒫 cpw 4600   cuni 4907  cima 5688  Oncon0 6384   Fn wfn 6556  wf 6557  cfv 6561   No csur 27684   M cmade 27881   O cold 27882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887
This theorem is referenced by:  oldssmade  27916  oldlim  27925  madebdayim  27926  oldbdayim  27927  madebdaylemold  27936
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