Theorem fsn2g 5830

Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)

Assertion

Ref	Expression
fsn2g	|- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )

Proof of Theorem fsn2g

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3684	. . 3 |- ( a = A -> { a } = { A } )
2	1	feq2d 5477	. 2 |- ( a = A -> ( F : { a } --> B <-> F : { A } --> B ) )
3		fveq2 5648	. . . 4 |- ( a = A -> ( F ` a ) = ( F ` A ) )
4	3	eleq1d 2300	. . 3 |- ( a = A -> ( ( F ` a ) e. B <-> ( F ` A ) e. B ) )
5		id 19	. . . . . 6 |- ( a = A -> a = A )
6	5, 3	opeq12d 3875	. . . . 5 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
7	6	sneqd 3686	. . . 4 |- ( a = A -> { <. a , ( F ` a ) >. } = { <. A , ( F ` A ) >. } )
8	7	eqeq2d 2243	. . 3 |- ( a = A -> ( F = { <. a , ( F ` a ) >. } <-> F = { <. A , ( F ` A ) >. } ) )
9	4, 8	anbi12d 473	. 2 |- ( a = A -> ( ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
10		vex 2806	. . 3 |- a e. _V
11	10	fsn2 5829	. 2 |- ( F : { a } --> B <-> ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) )
12	2, 9, 11	vtoclbg 2866	1 |- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {csn 3673   <.cop 3676   -->wf 5329   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341

This theorem is referenced by:  gsumsplit0  13996