Theorem fsn2 5873

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 5528	. . 3 |- ( F : { A } --> B -> F Fn { A } )
2		fsn2.1	. . . . 5 |- A e. _V
3	2	snid 3736	. . . 4 |- A e. { A }
4		funfvex 5707	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
5	4	funfni 5478	. . . 4 |- ( ( F Fn { A } /\ A e. { A } ) -> ( F ` A ) e. _V )
6	3, 5	mpan2 429	. . 3 |- ( F Fn { A } -> ( F ` A ) e. _V )
7	1, 6	syl 14	. 2 |- ( F : { A } --> B -> ( F ` A ) e. _V )
8		elex 2833	. . 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 5832	. . . . . 6 |- ( ( F : { A } --> B /\ A e. { A } ) -> ( F ` A ) e. B )
11	3, 10	mpan2 429	. . . . 5 |- ( F : { A } --> B -> ( F ` A ) e. B )
12		dffn3 5539	. . . . . . . 8 |- ( F Fn { A } <-> F : { A } --> ran F )
13	12	biimpi 120	. . . . . . 7 |- ( F Fn { A } -> F : { A } --> ran F )
14		imadmrn 5131	. . . . . . . . . 10 |- ( F " dom F ) = ran F
15		fndm 5475	. . . . . . . . . . 11 |- ( F Fn { A } -> dom F = { A } )
16	15	imaeq2d 5121	. . . . . . . . . 10 |- ( F Fn { A } -> ( F " dom F ) = ( F " { A } ) )
17	14, 16	eqtr3id 2285	. . . . . . . . 9 |- ( F Fn { A } -> ran F = ( F " { A } ) )
18		fnsnfv 5756	. . . . . . . . . 10 |- ( ( F Fn { A } /\ A e. { A } ) -> { ( F ` A ) } = ( F " { A } ) )
19	3, 18	mpan2 429	. . . . . . . . 9 |- ( F Fn { A } -> { ( F ` A ) } = ( F " { A } ) )
20	17, 19	eqtr4d 2274	. . . . . . . 8 |- ( F Fn { A } -> ran F = { ( F ` A ) } )
21		feq3 5513	. . . . . . . 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 3854	. . . . 5 |- ( ( F ` A ) e. B -> { ( F ` A ) } C_ B )
27		fss 5541	. . . . . 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 5872	. . . . 5 |- ( ( A e. _V /\ ( F ` A ) e. _V ) -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
32	2, 31	mpan 428	. . . 4 |- ( ( F ` A ) e. _V -> ( F : { A } --> { ( F ` A ) } <-> F = { <. A , ( F ` A ) >. } ) )
33	32	anbi2d 468	. . 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 716	1 |- ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {csn 3705   <.cop 3708   dom cdm 4769   ran crn 4770   "cima 4772   Fn wfn 5367   -->wf 5368   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by:  fsn2g  5874  fnressn  5892  fressnfv  5893  mapsnconst  6966  elixpsn  7007  en1  7076