Theorem fsn2g 5857

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 3705	. . 3 |- ( a = A -> { a } = { A } )
2	1	feq2d 5501	. 2 |- ( a = A -> ( F : { a } --> B <-> F : { A } --> B ) )
3		fveq2 5675	. . . 4 |- ( a = A -> ( F ` a ) = ( F ` A ) )
4	3	eleq1d 2303	. . 3 |- ( a = A -> ( ( F ` a ) e. B <-> ( F ` A ) e. B ) )
5		id 19	. . . . . 6 |- ( a = A -> a = A )
6	5, 3	opeq12d 3896	. . . . 5 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
7	6	sneqd 3707	. . . 4 |- ( a = A -> { <. a , ( F ` a ) >. } = { <. A , ( F ` A ) >. } )
8	7	eqeq2d 2246	. . 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 2818	. . 3 |- a e. _V
11	10	fsn2 5856	. 2 |- ( F : { a } --> B <-> ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) )
12	2, 9, 11	vtoclbg 2878	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 2205   {csn 3694   <.cop 3697   -->wf 5353   ` cfv 5357

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  gsumsplit0  14099