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Theorem dffv3 6887
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 6551 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 elimasng 6087 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 5149 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3bitr4di 288 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
54elvd 3481 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 6527 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
71, 6eqtr4id 2791 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8 fvprc 6883 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
9 snprc 4721 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110imaeq2d 6059 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12 ima0 6076 . . . . . . 7 (𝐹 “ ∅) = ∅
1311, 12eqtrdi 2788 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1413eleq2d 2819 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1514iotabidv 6527 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16 noel 4330 . . . . . . 7 ¬ 𝑥 ∈ ∅
1716nex 1802 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
18 euex 2571 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
1917, 18mto 196 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
20 iotanul 6521 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2119, 20ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2215, 21eqtrdi 2788 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
238, 22eqtr4d 2775 . 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 396   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2562  Vcvv 3474  c0 4322  {csn 4628  cop 4634   class class class wbr 5148  cima 5679  cio 6493  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551
This theorem is referenced by:  dffv4  6888  fvco2  6988  shftval  15020  dffv5  34891
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