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Theorem dffv3 6154
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 3193 . . . . 5 𝑥 ∈ V
2 elimasng 5460 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3 df-br 4624 . . . . . 6 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
42, 3syl6bbr 278 . . . . 5 ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
51, 4mpan2 706 . . . 4 (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
65iotabidv 5841 . . 3 (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7 df-fv 5865 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
86, 7syl6reqr 2674 . 2 (𝐴 ∈ V → (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
9 fvprc 6152 . . 3 𝐴 ∈ V → (𝐹𝐴) = ∅)
10 snprc 4230 . . . . . . . . 9 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 206 . . . . . . . 8 𝐴 ∈ V → {𝐴} = ∅)
1211imaeq2d 5435 . . . . . . 7 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
13 ima0 5450 . . . . . . 7 (𝐹 “ ∅) = ∅
1412, 13syl6eq 2671 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
1514eleq2d 2684 . . . . 5 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
1615iotabidv 5841 . . . 4 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
17 noel 3901 . . . . . . 7 ¬ 𝑥 ∈ ∅
1817nex 1728 . . . . . 6 ¬ ∃𝑥 𝑥 ∈ ∅
19 euex 2493 . . . . . 6 (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
2018, 19mto 188 . . . . 5 ¬ ∃!𝑥 𝑥 ∈ ∅
21 iotanul 5835 . . . . 5 (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
2220, 21ax-mp 5 . . . 4 (℩𝑥𝑥 ∈ ∅) = ∅
2316, 22syl6eq 2671 . . 3 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
249, 23eqtr4d 2658 . 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 384   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  Vcvv 3190  c0 3897  {csn 4155  cop 4161   class class class wbr 4623  cima 5087  cio 5818  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fv 5865
This theorem is referenced by:  dffv4  6155  fvco2  6240  shftval  13764  dffv5  31726
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