Theorem fsn2 4809

Description: A function that maps a singleton to a class is the singleton of an ordered pair.

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		fsn2.1	. . . . . 6 |- A e. _V
2	1	snid 3069	. . . . 5 |- A e. {A}
3		ffvelrn 4787	. . . . 5 |- ((F:{A}-->B /\ A e. {A}) -> (F` A) e. B)
4	2, 3	mpan2 760	. . . 4 |- (F:{A}-->B -> (F` A) e. B)
5		ffn 4562	. . . . 5 |- (F:{A}-->B -> F Fn {A})
6		dffn3 4570	. . . . . . 7 |- (F Fn {A} <-> F:{A}-->ran F)
7	6	biimpi 168	. . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8		fndm 4512	. . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
9	8	imaeq2d 4264	. . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10		imadmrn 4277	. . . . . . . . 9 |- (F"dom F) = ran F
11	9, 10	syl5eqr 1942	. . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12		fnsnfv 4728	. . . . . . . . 9 |- ((F Fn {A} /\ A e. {A}) -> {(F` A)} = (F"{A}))
13	2, 12	mpan2 760	. . . . . . . 8 |- (F Fn {A} -> {(F` A)} = (F"{A}))
14	11, 13	eqtr4d 1928	. . . . . . 7 |- (F Fn {A} -> ran F = {(F` A)})
15		feq3 4553	. . . . . . 7 |- (ran F = {(F` A)} -> (F:{A}-->ran F <-> F:{A}-->{(F` A)}))
16	14, 15	syl 12	. . . . . 6 |- (F Fn {A} -> (F:{A}-->ran F <-> F:{A}-->{(F` A)}))
17	7, 16	mpbid 212	. . . . 5 |- (F Fn {A} -> F:{A}-->{(F` A)})
18	5, 17	syl 12	. . . 4 |- (F:{A}-->B -> F:{A}-->{(F` A)})
19	4, 18	jca 310	. . 3 |- (F:{A}-->B -> ((F` A) e. B /\ F:{A}-->{(F` A)}))
20		fss 4571	. . . . 5 |- ((F:{A}-->{(F` A)} /\ {(F` A)} C_ B) -> F:{A}-->B)
21	20	ancoms 484	. . . 4 |- (({(F` A)} C_ B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
22		snssi 3129	. . . 4 |- ((F` A) e. B -> {(F` A)} C_ B)
23	21, 22	sylan 497	. . 3 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
24	19, 23	impbii 174	. 2 |- (F:{A}-->B <-> ((F` A) e. B /\ F:{A}-->{(F` A)}))
25		fvex 4689	. . . 4 |- (F` A) e. _V
26	1, 25	fsn 4807	. . 3 |- (F:{A}-->{(F` A)} <-> F = {<.A, (F` A)>.})
27	26	anbi2i 538	. 2 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
28	24, 27	bitri 190	1 |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  {csn 3044  <.cop 3046  dom cdm 3986  ran crn 3987  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  fnressn 4812  fressnfv 4813  en1 5485  bnj134 12478

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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