Theorem fsng 5691

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 3605	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 5355	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 3780	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 3607	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2189	. . 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 3605	. . . 4 |- ( b = B -> { b } = { B } )
8		feq3 5352	. . . 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 3781	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
11	10	sneqd 3607	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
12	11	eqeq2d 2189	. . 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 2742	. . 3 |- a e. _V
15		vex 2742	. . 3 |- b e. _V
16	14, 15	fsn 5690	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
17	6, 13, 16	vtocl2g 2803	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {csn 3594   <.cop 3597   -->wf 5214

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  fsn2  5692  xpsng  5693  ftpg  5702  fseq1p1m1  10096  intopsn  12791  grp1inv  12982