Theorem fsng 5658

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 3587	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 5325	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 3758	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 3589	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2177	. . 3 |- ( a = A -> ( F = { <. a , b >. } <-> F = { <. A , b >. } ) )
6	2, 5	bibi12d 234	. 2 |- ( a = A -> ( ( F : { a } --> { b } <-> F = { <. a , b >. } ) <-> ( F : { A } --> { b } <-> F = { <. A , b >. } ) ) )
7		sneq 3587	. . . 4 |- ( b = B -> { b } = { B } )
8		feq3 5322	. . . 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 3759	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
11	10	sneqd 3589	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
12	11	eqeq2d 2177	. . 3 |- ( b = B -> ( F = { <. A , b >. } <-> F = { <. A , B >. } ) )
13	9, 12	bibi12d 234	. 2 |- ( b = B -> ( ( F : { A } --> { b } <-> F = { <. A , b >. } ) <-> ( F : { A } --> { B } <-> F = { <. A , B >. } ) ) )
14		vex 2729	. . 3 |- a e. _V
15		vex 2729	. . 3 |- b e. _V
16	14, 15	fsn 5657	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
17	6, 13, 16	vtocl2g 2790	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {csn 3576   <.cop 3579   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  fsn2  5659  xpsng  5660  ftpg  5669  fseq1p1m1  10029  intopsn  12598