Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oldval Structured version   Visualization version   GIF version

Theorem oldval 33679
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 33672 . . . . 5 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
2 recsfnon 8068 . . . . . . 7 recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) Fn On
3 fnfun 6438 . . . . . . 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 6359 . . . . . 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 261 . . . . 5 ( M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))) → Fun M )
71, 6ax-mp 5 . . . 4 Fun M
8 funimaexg 6425 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
97, 8mpan 690 . . 3 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
109uniexd 7486 . 2 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
11 imaeq2 5899 . . . 4 (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
1211unieqd 4810 . . 3 (𝑥 = 𝐴 ( M “ 𝑥) = ( M “ 𝐴))
13 df-old 33673 . . 3 O = (𝑥 ∈ On ↦ ( M “ 𝑥))
1412, 13fvmptg 6773 . 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 1542  wcel 2114  Vcvv 3398  𝒫 cpw 4488   cuni 4796  cmpt 5110   × cxp 5523  ran crn 5526  cima 5528  Oncon0 6172  Fun wfun 6333   Fn wfn 6334  cfv 6339  recscrecs 8036   |s cscut 33618   M cmade 33667   O cold 33668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-wrecs 7976  df-recs 8037  df-made 33672  df-old 33673
This theorem is referenced by:  elmade2  33689  elold  33690  old0  33694  madeoldsuc  33705
  Copyright terms: Public domain W3C validator