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Theorem oldval 27893
Description: The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
Assertion
Ref Expression
oldval (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))

Proof of Theorem oldval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-made 27886 . . . . 5 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
2 recsfnon 8443 . . . . . . 7 recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On
3 fnfun 6668 . . . . . . 7 (recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On → Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))))
42, 3ax-mp 5 . . . . . 6 Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
5 funeq 6586 . . . . . 6 ( M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) → (Fun M ↔ Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))))
64, 5mpbiri 258 . . . . 5 ( M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) → Fun M )
71, 6ax-mp 5 . . . 4 Fun M
8 funimaexg 6653 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
97, 8mpan 690 . . 3 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
109uniexd 7762 . 2 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
11 imaeq2 6074 . . . 4 (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
1211unieqd 4920 . . 3 (𝑥 = 𝐴 ( M “ 𝑥) = ( M “ 𝐴))
13 df-old 27887 . . 3 O = (𝑥 ∈ On ↦ ( M “ 𝑥))
1412, 13fvmptg 7014 . 2 ((𝐴 ∈ On ∧ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ( M “ 𝐴))
1510, 14mpdan 687 1 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  𝒫 cpw 4600   cuni 4907  cmpt 5225   × cxp 5683  ran crn 5686  cima 5688  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  cfv 6561  recscrecs 8410   |s cscut 27827   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-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-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-op 4633  df-uni 4908  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-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-made 27886  df-old 27887
This theorem is referenced by:  old0  27898  elmade2  27907  elold  27908  old1  27914  madeoldsuc  27923  oldfi  27951
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