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Theorem dffv3 6897
Description: A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dffv3 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem dffv3
StepHypRef Expression
1 df-fv 6562 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 elimasng 6098 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 5154 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3bitr4di 288 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
54elvd 3469 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 6538 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
71, 6eqtr4id 2785 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8 fvprc 6893 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
9 snprc 4726 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110imaeq2d 6069 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12 ima0 6086 . . . . . . 7 (𝐹 “ ∅) = ∅
1311, 12eqtrdi 2782 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1413eleq2d 2812 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1514iotabidv 6538 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16 noel 4333 . . . . . . 7 ¬ 𝑥 ∈ ∅
1716nex 1795 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
18 euex 2566 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
1917, 18mto 196 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
20 iotanul 6532 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2119, 20ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2215, 21eqtrdi 2782 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
238, 22eqtr4d 2769 . 2 𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
247, 23pm2.61i 182 1 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2557  Vcvv 3462  c0 4325  {csn 4633  cop 4639   class class class wbr 5153  cima 5685  cio 6504  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562
This theorem is referenced by:  dffv4  6898  fvco2  6999  shftval  15079  dffv5  35748
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