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

Theorem dffv3 6336
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 vex 3331 . . . . 5 𝑥 ∈ V
2 elimasng 5637 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 4793 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3syl6bbr 278 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
51, 4mpan2 709 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 6021 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7 df-fv 6045 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
86, 7syl6reqr 2801 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
9 fvprc 6334 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
10 snprc 4385 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 206 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1211imaeq2d 5612 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
13 ima0 5627 . . . . . . 7 (𝐹 “ ∅) = ∅
1412, 13syl6eq 2798 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1514eleq2d 2813 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1615iotabidv 6021 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
17 noel 4050 . . . . . . 7 ¬ 𝑥 ∈ ∅
1817nex 1868 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
19 euex 2619 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
2018, 19mto 188 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
21 iotanul 6015 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2220, 21ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2316, 22syl6eq 2798 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
249, 23eqtr4d 2785 . 2 𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
258, 24pm2.61i 176 1 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1620  wex 1841  wcel 2127  ∃!weu 2595  Vcvv 3328  c0 4046  {csn 4309  cop 4315   class class class wbr 4792  cima 5257  cio 5998  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fv 6045
This theorem is referenced by:  dffv4  6337  fvco2  6423  shftval  13984  dffv5  32308
  Copyright terms: Public domain W3C validator