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Theorem dffv3g 5533
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 df-fv 5246 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 vex 2755 . . . 4 𝑥 ∈ V
3 elimasng 5017 . . . . 5 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
4 df-br 4022 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
53, 4bitr4di 198 . . . 4 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
62, 5mpan2 425 . . 3 (𝐴𝑉 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
76iotabidv 5221 . 2 (𝐴𝑉 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
81, 7eqtr4id 2241 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3610  cop 3613   class class class wbr 4021  cima 4650  cio 5197  cfv 5238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-cnv 4655  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fv 5246
This theorem is referenced by:  dffv4g  5534  fvco2  5609  shftval  10875
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