Theorem fsn2 5856

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 5513	. . 3 |- ( F : { A } --> B -> F Fn { A } )
2		fsn2.1	. . . . 5 |- A e. _V
3	2	snid 3725	. . . 4 |- A e. { A }
4		funfvex 5692	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
5	4	funfni 5463	. . . 4 |- ( ( F Fn { A } /\ A e. { A } ) -> ( F ` A ) e. _V )
6	3, 5	mpan2 425	. . 3 |- ( F Fn { A } -> ( F ` A ) e. _V )
7	1, 6	syl 14	. 2 |- ( F : { A } --> B -> ( F ` A ) e. _V )
8		elex 2827	. . 3 |- ( ( F ` A ) e. B -> ( F ` A ) e. _V )
9	8	adantr 276	. 2 |- ( ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) -> ( F ` A ) e. _V )
10		ffvelcdm 5815	. . . . . 6 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
11	3, 10	mpan2 425	. . . . 5 |- ( F : { A } --> B -> ( F ` A ) e. B )
12		dffn3 5524	. . . . . . . 8 |- ( F Fn { A } <-> F : { A } --> ran F )
13	12	biimpi 120	. . . . . . 7 |- ( F Fn { A } -> F : { A } --> ran F )
14		imadmrn 5116	. . . . . . . . . 10 |- ( F " dom F ) = ran F
15		fndm 5460	. . . . . . . . . . 11 |- ( F Fn { A } -> dom F = { A } )
16	15	imaeq2d 5106	. . . . . . . . . 10 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
17	14, 16	eqtr3id 2281	. . . . . . . . 9 |- ( F Fn { A } -> ran F = ( F " { A } ) )
18		fnsnfv 5741	. . . . . . . . . 10 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
19	3, 18	mpan2 425	. . . . . . . . 9 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
20	17, 19	eqtr4d 2270	. . . . . . . 8 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
21		feq3 5498	. . . . . . . 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 147	. . . . . 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 306	. . . 4 |- ( F : { A } --> B -> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
26		snssi 3843	. . . . 5 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
27		fss 5526	. . . . . 6 |- ( ( F : { A } --> { ( F ` A ) } /\ { ( F ` A ) } C_ B ) -> F : { A } --> B )
28	27	ancoms 268	. . . . 5 |- ( ( { ( F ` A ) } C_ B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
29	26, 28	sylan 283	. . . 4 |- ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) -> F : { A } --> B )
30	25, 29	impbii 126	. . 3 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) )
31		fsng 5855	. . . . 5 |- ( ( A e. _V /\ ( F ` A ) e. _V ) -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
32	2, 31	mpan 424	. . . 4 |- ( ( F ` A ) e. _V -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
33	32	anbi2d 464	. . 3 |- ( ( F ` A ) e. _V -> ( ( ( F ` A ) e. B /\ F : { A } --> { ( F ` A ) } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
34	30, 33	bitrid 192	. 2 |- ( ( F ` A ) e. _V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
35	7, 9, 34	pm5.21nii 712	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   {csn 3694   <.cop 3697   dom cdm 4754   ran crn 4755   "cima 4757   Fn wfn 5352   -->wf 5353   ` cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  fsn2g  5857  fnressn  5875  fressnfv  5876  mapsnconst  6942  elixpsn  6983  en1  7052