Theorem fsn2 5337

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 5046	. . 3 |- ( F : { A } --> B -> F Fn { A } )
2		fsn2.1	. . . . 5 |- A e. _V
3	2	snid 3402	. . . 4 |- A e. { A }
4		funfvex 5192	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
5	4	funfni 4999	. . . 4 |- ( ( F Fn { A } /\ A e. { A } ) -> ( F ` A ) e. _V )
6	3, 5	mpan2 401	. . 3 |- ( F Fn { A } -> ( F ` A ) e. _V )
7	1, 6	syl 14	. 2 |- ( F : { A } --> B -> ( F ` A ) e. _V )
8		elex 2566	. . 3 |- ( ( F ` A ) e. B -> ( F ` A ) e. _V )
9	8	adantr 261	. 2 |- ( ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) -> ( F ` A ) e. _V )
10		ffvelrn 5300	. . . . . 6 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
11	3, 10	mpan2 401	. . . . 5 |- ( F : { A } --> B -> ( F ` A ) e. B )
12		dffn3 5053	. . . . . . . 8 |- ( F Fn { A } <-> F : { A } --> ran F )
13	12	biimpi 113	. . . . . . 7 |- ( F Fn { A } -> F : { A } --> ran F )
14		imadmrn 4678	. . . . . . . . . 10 |- ( F " dom F ) = ran F
15		fndm 4998	. . . . . . . . . . 11 |- ( F Fn { A } -> dom F = { A } )
16	15	imaeq2d 4668	. . . . . . . . . 10 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
17	14, 16	syl5eqr 2086	. . . . . . . . 9 |- ( F Fn { A } -> ran F = ( F " { A } ) )
18		fnsnfv 5232	. . . . . . . . . 10 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
19	3, 18	mpan2 401	. . . . . . . . 9 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
20	17, 19	eqtr4d 2075	. . . . . . . 8 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
21		feq3 5032	. . . . . . . 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 135	. . . . . 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 290	. . . 4 |- ( F : { A } --> B -> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
26		snssi 3508	. . . . 5 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
27		fss 5054	. . . . . 6 |- ( ( F : { A } --> { ( F ` A ) } /\ { ( F ` A ) } C_ B ) -> F : { A } --> B )
28	27	ancoms 255	. . . . 5 |- ( ( { ( F ` A ) } C_ B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
29	26, 28	sylan 267	. . . 4 |- ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
30	25, 29	impbii 117	. . 3 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
31		fsng 5336	. . . . 5 |- ( ( A e. _V /\ ( F ` A ) e. _V ) -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
32	2, 31	mpan 400	. . . 4 |- ( ( F ` A ) e. _V -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
33	32	anbi2d 437	. . 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 181	. 2 |- ( ( F ` A ) e. _V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
35	7, 9, 34	pm5.21nii 620	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375   <.cop 3378   dom cdm 4345   ran crn 4346   "cima 4348   Fn wfn 4897   -->wf 4898   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by:  fnressn  5349  fressnfv  5350  en1  6279