Theorem fsng 5872

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 3716	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 5516	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 3899	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 3718	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2250	. . 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 3716	. . . 4 |- ( b = B -> { b } = { B } )
8		feq3 5513	. . . 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 3900	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
11	10	sneqd 3718	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
12	11	eqeq2d 2250	. . 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 2824	. . 3 |- a e. _V
15		vex 2824	. . 3 |- b e. _V
16	14, 15	fsn 5871	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
17	6, 13, 16	vtocl2g 2887	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379

This theorem is referenced by:  fsn2  5873  xpsng  5875  ftpg  5890  mapsnd  6960  fseq1p1m1  10479  cats1un  11471  intopsn  13664  grp1inv  13889