Theorem fsn2 5659

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 5337	. . 3 |- ( F : { A } --> B -> F Fn { A } )
2		fsn2.1	. . . . 5 |- A e. _V
3	2	snid 3607	. . . 4 |- A e. { A }
4		funfvex 5503	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
5	4	funfni 5288	. . . 4 |- ( ( F Fn { A } /\ A e. { A } ) -> ( F ` A ) e. _V )
6	3, 5	mpan2 422	. . 3 |- ( F Fn { A } -> ( F ` A ) e. _V )
7	1, 6	syl 14	. 2 |- ( F : { A } --> B -> ( F ` A ) e. _V )
8		elex 2737	. . 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 5618	. . . . . 6 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
11	3, 10	mpan2 422	. . . . 5 |- ( F : { A } --> B -> ( F ` A ) e. B )
12		dffn3 5348	. . . . . . . 8 |- ( F Fn { A } <-> F : { A } --> ran F )
13	12	biimpi 119	. . . . . . 7 |- ( F Fn { A } -> F : { A } --> ran F )
14		imadmrn 4956	. . . . . . . . . 10 |- ( F " dom F ) = ran F
15		fndm 5287	. . . . . . . . . . 11 |- ( F Fn { A } -> dom F = { A } )
16	15	imaeq2d 4946	. . . . . . . . . 10 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
17	14, 16	eqtr3id 2213	. . . . . . . . 9 |- ( F Fn { A } -> ran F = ( F " { A } ) )
18		fnsnfv 5545	. . . . . . . . . 10 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
19	3, 18	mpan2 422	. . . . . . . . 9 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
20	17, 19	eqtr4d 2201	. . . . . . . 8 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
21		feq3 5322	. . . . . . . 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 3717	. . . . 5 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
27		fss 5349	. . . . . 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 5658	. . . . 5 |- ( ( A e. _V /\ ( F ` A ) e. _V ) -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
32	2, 31	mpan 421	. . . 4 |- ( ( F ` A ) e. _V -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
33	32	anbi2d 460	. . 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 694	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {csn 3576   <.cop 3579   dom cdm 4604   ran crn 4605   "cima 4607   Fn wfn 5183   -->wf 5184   ` cfv 5188

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-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  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-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  fnressn  5671  fressnfv  5672  mapsnconst  6660  elixpsn  6701  en1  6765