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Theorem dffv3g 5095
 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 2534 . . . 4 x V
2 elimasng 4616 . . . . 5 ((A 𝑉 x V) → (x (𝐹 “ {A}) ↔ ⟨A, x 𝐹))
3 df-br 3735 . . . . 5 (A𝐹x ↔ ⟨A, x 𝐹)
42, 3syl6bbr 187 . . . 4 ((A 𝑉 x V) → (x (𝐹 “ {A}) ↔ A𝐹x))
51, 4mpan2 403 . . 3 (A 𝑉 → (x (𝐹 “ {A}) ↔ A𝐹x))
65iotabidv 4811 . 2 (A 𝑉 → (℩xx (𝐹 “ {A})) = (℩xA𝐹x))
7 df-fv 4833 . 2 (𝐹A) = (℩xA𝐹x)
86, 7syl6reqr 2069 1 (A 𝑉 → (𝐹A) = (℩xx (𝐹 “ {A})))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  Vcvv 2531  {csn 3346  ⟨cop 3349   class class class wbr 3734   “ cima 4271  ℩cio 4788  ‘cfv 4825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fv 4833 This theorem is referenced by:  dffv4g  5096  fvco2  5163
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