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Theorem oldval 27908
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 27901 . . . . 5 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
2 recsfnon 8442 . . . . . . 7 recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On
3 fnfun 6669 . . . . . . 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 6588 . . . . . 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 6654 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
97, 8mpan 690 . . 3 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
109uniexd 7761 . 2 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
11 imaeq2 6076 . . . 4 (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
1211unieqd 4925 . . 3 (𝑥 = 𝐴 ( M “ 𝑥) = ( M “ 𝐴))
13 df-old 27902 . . 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 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605   cuni 4912  cmpt 5231   × cxp 5687  ran crn 5690  cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  cfv 6563  recscrecs 8409   |s cscut 27842   M cmade 27896   O cold 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-made 27901  df-old 27902
This theorem is referenced by:  old0  27913  elmade2  27922  elold  27923  old1  27929  madeoldsuc  27938  oldfi  27966
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