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Theorem oldval 27187
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 27180 . . . . 5 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
2 recsfnon 8350 . . . . . . 7 recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On
3 fnfun 6603 . . . . . . 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 6522 . . . . . 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 6588 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
97, 8mpan 689 . . 3 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
109uniexd 7680 . 2 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
11 imaeq2 6010 . . . 4 (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
1211unieqd 4880 . . 3 (𝑥 = 𝐴 ( M “ 𝑥) = ( M “ 𝐴))
13 df-old 27181 . . 3 O = (𝑥 ∈ On ↦ ( M “ 𝑥))
1412, 13fvmptg 6947 . 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 3446  𝒫 cpw 4561   cuni 4866  cmpt 5189   × cxp 5632  ran crn 5635  cima 5637  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  cfv 6497  recscrecs 8317   |s cscut 27125   M cmade 27175   O cold 27176
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-made 27180  df-old 27181
This theorem is referenced by:  old0  27192  elmade2  27201  elold  27202  old1  27208  madeoldsuc  27217
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