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Theorem dffv3 6903
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 6571 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 elimasng 6109 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 5149 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3bitr4di 289 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
54elvd 3484 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 6547 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
71, 6eqtr4id 2794 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8 fvprc 6899 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
9 snprc 4722 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 216 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110imaeq2d 6080 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12 ima0 6097 . . . . . . 7 (𝐹 “ ∅) = ∅
1311, 12eqtrdi 2791 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1413eleq2d 2825 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1514iotabidv 6547 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16 noel 4344 . . . . . . 7 ¬ 𝑥 ∈ ∅
1716nex 1797 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
18 euex 2575 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
1917, 18mto 197 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
20 iotanul 6541 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2119, 20ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2215, 21eqtrdi 2791 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
238, 22eqtr4d 2778 . 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 206  wa 395   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  Vcvv 3478  c0 4339  {csn 4631  cop 4637   class class class wbr 5148  cima 5692  cio 6514  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571
This theorem is referenced by:  dffv4  6904  fvco2  7006  shftval  15110  dffv5  35906
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