Theorem dffv5 36509

Description: Another quantifier-free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
dffv5	⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )

Proof of Theorem dffv5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffv3 6878	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2		dfiota3 36508	. 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3		abid2 2899	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	sneqi 4598	. . . . 5 ⊢ {{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
5	4	ineq1i 4165	. . . 4 ⊢ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
6	5	unieqi 4882	. . 3 ⊢ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )
7	6	unieqi 4882	. 2 ⊢ ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )
8	1, 2, 7	3eqtri 2789	1 ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2740   ∩ cin 3901  {csn 4587  ∪ cuni 4870   “ cima 5662  ℩cio 6491  ‘cfv 6537   Singletons csingles 36424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4202  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7990  df-2nd 7991  df-txp 36439  df-singleton 36447  df-singles 36448

This theorem is used by:  brapply  36523