Theorem dffv5 36894

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 6910	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2		dfiota3 36893	. 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3		abid2 2901	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	sneqi 4595	. . . . 5 ⊢ {{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
5	4	ineq1i 4116	. . . 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 2790	1 ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741   ∩ cin 4092  {csn 4588  ∪ cuni 4870   “ cima 5659  ℩cio 6516  ‘cfv 6564   Singletons csingles 36809

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-symdif 4201  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-eprel 5556  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-fo 6570  df-fv 6572  df-1st 8033  df-2nd 8034  df-txp 36824  df-singleton 36832  df-singles 36833

This theorem is used by:  brapply  36908