Theorem fsn2 3912

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 2480	. . . . 5 |- A e. {A}
3		ffvelrn 3890	. . . . 5 |- ((F:{A}-->B /\ A e. {A}) -> (F` A) e. B)
4	2, 3	mpan2 699	. . . 4 |- (F:{A}-->B -> (F` A) e. B)
5		ffn 3702	. . . . 5 |- (F:{A}-->B -> F Fn {A})
6		dffn3 3709	. . . . . . 7 |- (F Fn {A} <-> F:{A}-->ran F)
7	6	biimpi 149	. . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8		fndm 3662	. . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
9	8	imaeq2d 3467	. . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10		imadmrn 3477	. . . . . . . . 9 |- (F"dom F) = ran F
11	9, 10	syl5eqr 1558	. . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12		fnsnfv 3843	. . . . . . . . 9 |- ((F Fn {A} /\ A e. {A}) -> {(F` A)} = (F"{A}))
13	2, 12	mpan2 699	. . . . . . . 8 |- (F Fn {A} -> {(F` A)} = (F"{A}))
14	11, 13	eqtr4d 1547	. . . . . . 7 |- (F Fn {A} -> ran F = {(F` A)})
15		feq3 3697	. . . . . . 7 |- (ran F = {(F` A)} -> (F:{A}-->ran F <-> F:{A}-->{(F` A)}))
16	14, 15	syl 10	. . . . . 6 |- (F Fn {A} -> (F:{A}-->ran F <-> F:{A}-->{(F` A)}))
17	7, 16	mpbid 193	. . . . 5 |- (F Fn {A} -> F:{A}-->{(F` A)})
18	5, 17	syl 10	. . . 4 |- (F:{A}-->B -> F:{A}-->{(F` A)})
19	4, 18	jca 286	. . 3 |- (F:{A}-->B -> ((F` A) e. B /\ F:{A}-->{(F` A)}))
20		fss 3710	. . . . 5 |- ((F:{A}-->{(F` A)} /\ {(F` A)} (_ B) -> F:{A}-->B)
21	20	ancoms 438	. . . 4 |- (({(F` A)} (_ B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
22		snssi 2514	. . . 4 |- ((F` A) e. B -> {(F` A)} (_ B)
23	21, 22	sylan 450	. . 3 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
24	19, 23	impbii 155	. 2 |- (F:{A}-->B <-> ((F` A) e. B /\ F:{A}-->{(F` A)}))
25		fvex 3808	. . . 4 |- (F` A) e. V
26	1, 25	fsn 3910	. . 3 |- (F:{A}-->{(F` A)} <-> F = {<.A, (F` A)>.})
27	26	anbi2i 482	. 2 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
28	24, 27	bitri 171	1 |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))

Syntax hints:   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  Vcvv 1849   (_ wss 2091  {csn 2454  <.cop 2456  dom cdm 3225  ran crn 3226  "cima 3228   Fn wfn 3232  -->wf 3233  ` cfv 3237

This theorem is referenced by:  fnressn 3913  fressnfv 3914  en1 4513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253

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