Theorem fsng 6404

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 4187	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 6031	. . 3 |- ( a = A -> ( F : { a } --> { b } <-> F : { A } --> { b } ) )
3		opeq1 4402	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
4	3	sneqd 4189	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
5	4	eqeq2d 2632	. . 3 |- ( a = A -> ( F = { <. a , b >. } <-> F = { <. A , b >. } ) )
6	2, 5	bibi12d 335	. 2 |- ( a = A -> ( ( F : { a } --> { b } <-> F = { <. a , b >. } ) <-> ( F : { A } --> { b } <-> F = { <. A , b >. } ) ) )
7		sneq 4187	. . . 4 |- ( b = B -> { b } = { B } )
8	7	feq3d 6032	. . 3 |- ( b = B -> ( F : { A } --> { b } <-> F : { A } --> { B } ) )
9		opeq2 4403	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
10	9	sneqd 4189	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
11	10	eqeq2d 2632	. . 3 |- ( b = B -> ( F = { <. A , b >. } <-> F = { <. A , B >. } ) )
12	8, 11	bibi12d 335	. 2 |- ( b = B -> ( ( F : { A } --> { b } <-> F = { <. A , b >. } ) <-> ( F : { A } --> { B } <-> F = { <. A , B >. } ) ) )
13		vex 3203	. . 3 |- a e. _V
14		vex 3203	. . 3 |- b e. _V
15	13, 14	fsn 6402	. 2 |- ( F : { a } --> { b } <-> F = { <. a , b >. } )
16	6, 12, 15	vtocl2g 3270	1 |- ( ( A e. C /\ B e. D ) -> ( F : { A } --> { B } <-> F = { <. A , B >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  xpsng  6406  ftpg  6423  axdc3lem4  9275  fseq1p1m1  12414  cats1un  13475  intopsn  17253  grp1inv  17523  symg1bas  17816  esumsnf  30126  bnj149  30945  rngosn3  33723  k0004lem3  38447  mapsnd  39388  ovnovollem1  40870  mapsnop  42123  snlindsntorlem  42259  lmod1zr  42282