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Theorem dffv5 32156
Description: Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dffv5 (𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )

Proof of Theorem dffv5
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dffv3 6225 . 2 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2 dfiota3 32155 . 2 (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3 abid2 2774 . . . . . 6 {𝑥𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43sneqi 4221 . . . . 5 {{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
54ineq1i 3843 . . . 4 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
65unieqi 4477 . . 3 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
76unieqi 4477 . 2 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
81, 2, 73eqtri 2677 1 (𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {cab 2637  cin 3606  {csn 4210   cuni 4468  cima 5146  cio 5887  cfv 5926   Singletons csingles 32071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-singleton 32094  df-singles 32095
This theorem is referenced by:  brapply  32170
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