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Theorem newval 27909
Description: The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
newval ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))

Proof of Theorem newval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴))
2 fveq2 6907 . . . 4 (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
31, 2difeq12d 4137 . . 3 (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
4 df-new 27903 . . 3 N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥)))
5 fvex 6920 . . . 4 ( M ‘𝐴) ∈ V
65difexi 5336 . . 3 (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V
73, 4, 6fvmpt 7016 . 2 (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
84fvmptndm 7047 . . 3 𝐴 ∈ On → ( N ‘𝐴) = ∅)
9 df-made 27901 . . . . . . . . 9 M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ran 𝑓 × 𝒫 ran 𝑓))))
109tfr1 8436 . . . . . . . 8 M Fn On
1110fndmi 6673 . . . . . . 7 dom M = On
1211eleq2i 2831 . . . . . 6 (𝐴 ∈ dom M ↔ 𝐴 ∈ On)
13 ndmfv 6942 . . . . . 6 𝐴 ∈ dom M → ( M ‘𝐴) = ∅)
1412, 13sylnbir 331 . . . . 5 𝐴 ∈ On → ( M ‘𝐴) = ∅)
1514difeq1d 4135 . . . 4 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴)))
16 0dif 4411 . . . 4 (∅ ∖ ( O ‘𝐴)) = ∅
1715, 16eqtrdi 2791 . . 3 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅)
188, 17eqtr4d 2778 . 2 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
197, 18pm2.61i 182 1 ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  c0 4339  𝒫 cpw 4605   cuni 4912  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cima 5692  Oncon0 6386  cfv 6563   |s cscut 27842   M cmade 27896   O cold 27897   N cnew 27898
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-new 27903
This theorem is referenced by:  new0  27928  madeun  27937  newbday  27955
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