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Theorem dffv3g 5336
Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dffv3g (𝐴𝑉 → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝑉

Proof of Theorem dffv3g
StepHypRef Expression
1 vex 2636 . . . 4 𝑥 ∈ V
2 elimasng 4833 . . . . 5 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 3868 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3syl6bbr 197 . . . 4 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
51, 4mpan2 417 . . 3 (𝐴𝑉 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 5035 . 2 (𝐴𝑉 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7 df-fv 5057 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
86, 7syl6reqr 2146 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wcel 1445  Vcvv 2633  {csn 3466  cop 3469   class class class wbr 3867  cima 4470  cio 5012  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fv 5057
This theorem is referenced by:  dffv4g  5337  fvco2  5408  shftval  10374
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