Theorem fsn2 6403

Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Hypothesis

Ref	Expression
fsn2.1	|- A e. _V

Assertion

Ref	Expression
fsn2	|- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Proof of Theorem fsn2

Step	Hyp	Ref	Expression
1		fsn2.1	. . . . . 6 |- A e. _V
2	1	snid 4208	. . . . 5 |- A e. { A }
3		ffvelrn 6357	. . . . 5 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
4	2, 3	mpan2 707	. . . 4 |- ( F : { A } --> B -> ( F ` A ) e. B )
5		ffn 6045	. . . . 5 |- ( F : { A } --> B -> F Fn { A } )
6		dffn3 6054	. . . . . . 7 |- ( F Fn { A } <-> F : { A } --> ran F )
7	6	biimpi 206	. . . . . 6 |- ( F Fn { A } -> F : { A } --> ran F )
8		imadmrn 5476	. . . . . . . . 9 |- ( F " dom F ) = ran F
9		fndm 5990	. . . . . . . . . 10 |- ( F Fn { A } -> dom F = { A } )
10	9	imaeq2d 5466	. . . . . . . . 9 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
11	8, 10	syl5eqr 2670	. . . . . . . 8 |- ( F Fn { A } -> ran F = ( F " { A } ) )
12		fnsnfv 6258	. . . . . . . . 9 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
13	2, 12	mpan2 707	. . . . . . . 8 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
14	11, 13	eqtr4d 2659	. . . . . . 7 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
15	14	feq3d 6032	. . . . . 6 |- ( F Fn { A } -> ( F : { A } --> ran F <-> F : { A } --> { ( F ` A ) } ) )
16	7, 15	mpbid 222	. . . . 5 |- ( F Fn { A } -> F : { A } --> { ( F ` A ) } )
17	5, 16	syl 17	. . . 4 |- ( F : { A } --> B -> F : { A } --> { ( F ` A ) } )
18	4, 17	jca 554	. . 3 |- ( F : { A } --> B -> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
19		snssi 4339	. . . 4 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
20		fss 6056	. . . . 5 |- ( ( F : { A } --> { ( F ` A ) } /\ { ( F ` A ) } C_ B ) -> F : { A } --> B )
21	20	ancoms 469	. . . 4 |- ( ( { ( F ` A ) } C_ B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
22	19, 21	sylan 488	. . 3 |- ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
23	18, 22	impbii 199	. 2 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
24		fvex 6201	. . . 4 |- ( F ` A ) e. _V
25	1, 24	fsn 6402	. . 3 |- ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } )
26	25	anbi2i 730	. 2 |- ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )
27	23, 26	bitri 264	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  fsn2g  6405  fnressn  6425  fressnfv  6427  mapsnconst  7903  elixpsn  7947  en1  8023  mat1dimelbas  20277  0spth  26987  ldepsnlinclem1  42294  ldepsnlinclem2  42295