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Theorem dffv3g 5117
Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dffv3g (A 𝑉 → (𝐹A) = (℩xx (𝐹 “ {A})))
Distinct variable groups:   x,𝐹   x,A   x,𝑉

Proof of Theorem dffv3g
StepHypRef Expression
1 vex 2554 . . . 4 x V
2 elimasng 4636 . . . . 5 ((A 𝑉 x V) → (x (𝐹 “ {A}) ↔ ⟨A, x 𝐹))
3 df-br 3756 . . . . 5 (A𝐹x ↔ ⟨A, x 𝐹)
42, 3syl6bbr 187 . . . 4 ((A 𝑉 x V) → (x (𝐹 “ {A}) ↔ A𝐹x))
51, 4mpan2 401 . . 3 (A 𝑉 → (x (𝐹 “ {A}) ↔ A𝐹x))
65iotabidv 4831 . 2 (A 𝑉 → (℩xx (𝐹 “ {A})) = (℩xA𝐹x))
7 df-fv 4853 . 2 (𝐹A) = (℩xA𝐹x)
86, 7syl6reqr 2088 1 (A 𝑉 → (𝐹A) = (℩xx (𝐹 “ {A})))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   class class class wbr 3755  cima 4291  cio 4808  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fv 4853
This theorem is referenced by:  dffv4g  5118  fvco2  5185
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