Theorem dffv5 36135

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 6838	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
2		dfiota3 36134	. 2 ⊢ (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons )
3		abid2 2874	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	sneqi 4593	. . . . 5 ⊢ {{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} = {(𝐹 “ {𝐴})}
5	4	ineq1i 4170	. . . 4 ⊢ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ({(𝐹 “ {𝐴})} ∩ Singletons )
6	5	unieqi 4877	. . 3 ⊢ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )
7	6	unieqi 4877	. 2 ⊢ ∪ ∪ ({{𝑥 ∣ 𝑥 ∈ (𝐹 “ {𝐴})}} ∩ Singletons ) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )
8	1, 2, 7	3eqtri 2764	1 ⊢ (𝐹‘𝐴) = ∪ ∪ ({(𝐹 “ {𝐴})} ∩ Singletons )

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2715   ∩ cin 3902  {csn 4582  ∪ cuni 4865   “ cima 5635  ℩cio 6454  ‘cfv 6500   Singletons csingles 36050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36065  df-singleton 36073  df-singles 36074

This theorem is referenced by:  brapply  36149