MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffv3 Structured version   Visualization version   GIF version

Theorem dffv3 6662
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 elimasng 5952 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
2 df-br 5063 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
31, 2syl6bbr 290 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
43elvd 3505 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
54iotabidv 6336 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
6 df-fv 6359 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
75, 6syl6reqr 2879 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8 fvprc 6659 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
9 snprc 4651 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 217 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1110imaeq2d 5926 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12 ima0 5942 . . . . . . 7 (𝐹 “ ∅) = ∅
1311, 12syl6eq 2876 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1413eleq2d 2902 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1514iotabidv 6336 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16 noel 4299 . . . . . . 7 ¬ 𝑥 ∈ ∅
1716nex 1794 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
18 euex 2659 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
1917, 18mto 198 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
20 iotanul 6330 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2119, 20ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2215, 21syl6eq 2876 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
238, 22eqtr4d 2863 . 2 𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
247, 23pm2.61i 183 1 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  ∃!weu 2650  Vcvv 3499  c0 4294  {csn 4563  cop 4569   class class class wbr 5062  cima 5556  cio 6309  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fv 6359
This theorem is referenced by:  dffv4  6663  fvco2  6754  shftval  14426  dffv5  33269
  Copyright terms: Public domain W3C validator