Theorem fsng 5855

Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Assertion

Ref	Expression
fsng	|- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Proof of Theorem fsng

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3705	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 5501	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 3888	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 3707	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2246	. . 3 |- ( a = A -> ( F = { <. a , b >. } <-> F = { <. A , b >. } ) )
6	2, 5	bibi12d 235	. 2 |- ( a = A -> ( ( F : { a } --> { b } <-> F = { <. a , b >. } ) <-> ( F : { A } --> { b } <-> F = { <. A , b >. } ) ) )
7		sneq 3705	. . . 4 |- ( b = B -> { b } = { B } )
8		feq3 5498	. . . 4 |- ( { b } = { B } -> ( F : { A } --> { b } <-> F : { A } --> { B } ) )
9	7, 8	syl 14	. . 3 |- ( b = B -> ( F : { A } --> { b } <-> F : { A } --> { B } ) )
10		opeq2 3889	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
11	10	sneqd 3707	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
12	11	eqeq2d 2246	. . 3 |- ( b = B -> ( F = { <. A , b >. } <-> F = { <. A , B >. } ) )
13	9, 12	bibi12d 235	. 2 |- ( b = B -> ( ( F : { A } --> { b } <-> F = { <. A , b >. } ) <-> ( F : { A } --> { B } <-> F = { <. A , B >. } ) ) )
14		vex 2818	. . 3 |- a e. _V
15		vex 2818	. . 3 |- b e. _V
16	14, 15	fsn 5854	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
17	6, 13, 16	vtocl2g 2881	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {csn 3694   <.cop 3697   -->wf 5353

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-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  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-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by:  fsn2  5856  xpsng  5858  ftpg  5873  mapsnd  6936  fseq1p1m1  10450  cats1un  11438  intopsn  13630  grp1inv  13862