Theorem fsng 5776

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 3654	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 5433	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 3833	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 3656	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2219	. . 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 3654	. . . 4 |- ( b = B -> { b } = { B } )
8		feq3 5430	. . . 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 3834	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
11	10	sneqd 3656	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
12	11	eqeq2d 2219	. . 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 2779	. . 3 |- a e. _V
15		vex 2779	. . 3 |- b e. _V
16	14, 15	fsn 5775	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
17	6, 13, 16	vtocl2g 2842	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   {csn 3643   <.cop 3646   -->wf 5286

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297

This theorem is referenced by:  fsn2  5777  xpsng  5778  ftpg  5791  fseq1p1m1  10251  cats1un  11212  intopsn  13314  grp1inv  13554