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Theorem dffv3g 5137
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 2557 . . . 4 𝑥 ∈ V
2 elimasng 4656 . . . . 5 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 3762 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3syl6bbr 187 . . . 4 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
51, 4mpan2 401 . . 3 (𝐴𝑉 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 4851 . 2 (𝐴𝑉 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7 df-fv 4873 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
86, 7syl6reqr 2091 1 (𝐴𝑉 → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  Vcvv 2554  {csn 3372  cop 3375   class class class wbr 3761  cima 4311  cio 4828  cfv 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-xp 4314  df-cnv 4316  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fv 4873
This theorem is referenced by:  dffv4g  5138  fvco2  5205  shftval  9280
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