Theorem fsn2 5587

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		ffn 5267	. . 3 |- ( F : { A } --> B -> F Fn { A } )
2		fsn2.1	. . . . 5 |- A e. _V
3	2	snid 3551	. . . 4 |- A e. { A }
4		funfvex 5431	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
5	4	funfni 5218	. . . 4 |- ( ( F Fn { A } /\ A e. { A } ) -> ( F ` A ) e. _V )
6	3, 5	mpan2 421	. . 3 |- ( F Fn { A } -> ( F ` A ) e. _V )
7	1, 6	syl 14	. 2 |- ( F : { A } --> B -> ( F ` A ) e. _V )
8		elex 2692	. . 3 |- ( ( F ` A ) e. B -> ( F ` A ) e. _V )
9	8	adantr 274	. 2 |- ( ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) -> ( F ` A ) e. _V )
10		ffvelrn 5546	. . . . . 6 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
11	3, 10	mpan2 421	. . . . 5 |- ( F : { A } --> B -> ( F ` A ) e. B )
12		dffn3 5278	. . . . . . . 8 |- ( F Fn { A } <-> F : { A } --> ran F )
13	12	biimpi 119	. . . . . . 7 |- ( F Fn { A } -> F : { A } --> ran F )
14		imadmrn 4886	. . . . . . . . . 10 |- ( F " dom F ) = ran F
15		fndm 5217	. . . . . . . . . . 11 |- ( F Fn { A } -> dom F = { A } )
16	15	imaeq2d 4876	. . . . . . . . . 10 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
17	14, 16	syl5eqr 2184	. . . . . . . . 9 |- ( F Fn { A } -> ran F = ( F " { A } ) )
18		fnsnfv 5473	. . . . . . . . . 10 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
19	3, 18	mpan2 421	. . . . . . . . 9 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
20	17, 19	eqtr4d 2173	. . . . . . . 8 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
21		feq3 5252	. . . . . . . 8 |- ( ran F = { ( F ` A ) } -> ( F : { A } --> ran F <-> F : { A } --> { ( F ` A ) } ) )
22	20, 21	syl 14	. . . . . . 7 |- ( F Fn { A } -> ( F : { A } --> ran F <-> F : { A } --> { ( F ` A ) } ) )
23	13, 22	mpbid 146	. . . . . 6 |- ( F Fn { A } -> F : { A } --> { ( F ` A ) } )
24	1, 23	syl 14	. . . . 5 |- ( F : { A } --> B -> F : { A } --> { ( F ` A ) } )
25	11, 24	jca 304	. . . 4 |- ( F : { A } --> B -> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
26		snssi 3659	. . . . 5 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
27		fss 5279	. . . . . 6 |- ( ( F : { A } --> { ( F ` A ) } /\ { ( F ` A ) } C_ B ) -> F : { A } --> B )
28	27	ancoms 266	. . . . 5 |- ( ( { ( F ` A ) } C_ B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
29	26, 28	sylan 281	. . . 4 |- ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
30	25, 29	impbii 125	. . 3 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
31		fsng 5586	. . . . 5 |- ( ( A e. _V /\ ( F ` A ) e. _V ) -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
32	2, 31	mpan 420	. . . 4 |- ( ( F ` A ) e. _V -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
33	32	anbi2d 459	. . 3 |- ( ( F ` A ) e. _V -> ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
34	30, 33	syl5bb 191	. 2 |- ( ( F ` A ) e. _V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
35	7, 9, 34	pm5.21nii 693	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2681   C_ wss 3066   {csn 3522   <.cop 3525   dom cdm 4534   ran crn 4535   "cima 4537   Fn wfn 5113   -->wf 5114   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126

This theorem is referenced by:  fnressn  5599  fressnfv  5600  mapsnconst  6581  elixpsn  6622  en1  6686