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Theorem dffv5 34538
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 6843 . 2 (𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2 dfiota3 34537 . 2 (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3 abid2 2876 . . . . . 6 {𝑥𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43sneqi 4602 . . . . 5 {{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
54ineq1i 4173 . . . 4 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
65unieqi 4883 . . 3 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
76unieqi 4883 . 2 ({{𝑥𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
81, 2, 73eqtri 2769 1 (𝐹𝐴) = ({(𝐹 “ {𝐴})} ∩ Singletons )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2714  cin 3914  {csn 4591   cuni 4870  cima 5641  cio 6451  cfv 6501   Singletons csingles 34453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-symdif 4207  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-eprel 5542  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fo 6507  df-fv 6509  df-1st 7926  df-2nd 7927  df-txp 34468  df-singleton 34476  df-singles 34477
This theorem is referenced by:  brapply  34552
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