Theorem fsn2g 5854

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 3702	. . 3 |- ( a = A -> { a } = { A } )
2	1	feq2d 5498	. 2 |- ( a = A -> ( F : { a } --> B <-> F : { A } --> B ) )
3		fveq2 5672	. . . 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 3893	. . . . 5 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
7	6	sneqd 3704	. . . 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 5853	. 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 3691   <.cop 3694   -->wf 5350   ` cfv 5354

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 4230  ax-pow 4289  ax-pr 4324

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 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362

This theorem is referenced by:  gsumsplit0  14080