MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oldval Structured version   Visualization version   GIF version

Theorem oldval 27339
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 27332 . . . . 5 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
2 recsfnon 8400 . . . . . . 7 recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On
3 fnfun 6647 . . . . . . 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 6566 . . . . . 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 6632 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
97, 8mpan 689 . . 3 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
109uniexd 7729 . 2 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
11 imaeq2 6054 . . . 4 (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
1211unieqd 4922 . . 3 (𝑥 = 𝐴 ( M “ 𝑥) = ( M “ 𝐴))
13 df-old 27333 . . 3 O = (𝑥 ∈ On ↦ ( M “ 𝑥))
1412, 13fvmptg 6994 . 2 ((𝐴 ∈ On ∧ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ( M “ 𝐴))
1510, 14mpdan 686 1 (𝐴 ∈ On → ( O ‘𝐴) = ( M “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  𝒫 cpw 4602   cuni 4908  cmpt 5231   × cxp 5674  ran crn 5677  cima 5679  Oncon0 6362  Fun wfun 6535   Fn wfn 6536  cfv 6541  recscrecs 8367   |s cscut 27274   M cmade 27327   O cold 27328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-made 27332  df-old 27333
This theorem is referenced by:  old0  27344  elmade2  27353  elold  27354  old1  27360  madeoldsuc  27369
  Copyright terms: Public domain W3C validator