Theorem fsn2 3842

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 2439	. . . . 5 |- A e. {A}
3		ffvelrn 3820	. . . . 5 |- ((F:{A}-->B /\ A e. {A}) -> (F` A) e. B)
4	2, 3	mpan2 698	. . . 4 |- (F:{A}-->B -> (F` A) e. B)
5		ffn 3633	. . . . 5 |- (F:{A}-->B -> F Fn {A})
6		fnfrn 3640	. . . . . . 7 |- (F Fn {A} <-> F:{A}-->ran F)
7	6	biimp 151	. . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8		fndm 3593	. . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
9	8	imaeq2d 3410	. . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10		imadmrn 3420	. . . . . . . . 9 |- (F"dom F) = ran F
11	9, 10	syl5eqr 1524	. . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12		fnsnfv 3773	. . . . . . . . 9 |- ((F Fn {A} /\ A e. {A}) -> {(F` A)} = (F"{A}))
13	2, 12	mpan2 698	. . . . . . . 8 |- (F Fn {A} -> {(F` A)} = (F"{A}))
14	11, 13	eqtr4d 1513	. . . . . . 7 |- (F Fn {A} -> ran F = {(F` A)})
15		feq3 3628	. . . . . . 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 195	. . . . 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 288	. . 3 |- (F:{A}-->B -> ((F` A) e. B /\ F:{A}-->{(F` A)}))
20		fss 3641	. . . . 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 2470	. . . 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	impbi 157	. 2 |- (F:{A}-->B <-> ((F` A) e. B /\ F:{A}-->{(F` A)}))
25		fvex 3738	. . . 4 |- (F` A) e. V
26	1, 25	fsn 3840	. . 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	bitr 173	1 |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   (_ wss 2050  {csn 2413  <.cop 2415  dom cdm 3176  ran crn 3177  "cima 3179   Fn wfn 3183  -->wf 3184  ` cfv 3188

This theorem is referenced by:  fnressn 3843  fressnfv 3844  en1 4432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-reu 1654  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204

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