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Theorem dffv5 35905
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 6902 . 2 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2 dfiota3 35904 . 2 (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3 abid2 2876 . . . . . 6 {𝑥𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43sneqi 4641 . . . . 5 {{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
54ineq1i 4223 . . . 4 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
65unieqi 4923 . . 3 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
76unieqi 4923 . 2 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
81, 2, 73eqtri 2766 1 (𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  {cab 2711  cin 3961  {csn 4630   cuni 4911  cima 5691  cio 6513  cfv 6562   Singletons csingles 35820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-symdif 4258  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-singleton 35843  df-singles 35844
This theorem is referenced by:  brapply  35919
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