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Theorem dffv5 30994
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 6083 . 2 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2 dfiota3 30993 . 2 (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3 abid2 2731 . . . . . 6 {𝑥𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43sneqi 4135 . . . . 5 {{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
54ineq1i 3771 . . . 4 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
65unieqi 4375 . . 3 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
76unieqi 4375 . 2 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
81, 2, 73eqtri 2635 1 (𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  {cab 2595  cin 3538  {csn 4124   cuni 4366  cima 5030  cio 5751  cfv 5789   Singletons csingles 30908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-symdif 3805  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-eprel 4938  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-fo 5795  df-fv 5797  df-1st 7036  df-2nd 7037  df-txp 30923  df-singleton 30931  df-singles 30932
This theorem is referenced by:  brapply  31008
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