Theorem fsn2 3831

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 2432	. . . . 5 |- A e. {A}
3		ffvelrn 3809	. . . . 5 |- ((F:{A}-->B /\ A e. {A}) -> (F` A) e. B)
4	2, 3	mpan2 695	. . . 4 |- (F:{A}-->B -> (F` A) e. B)
5		ffn 3623	. . . . 5 |- (F:{A}-->B -> F Fn {A})
6		fnfrn 3629	. . . . . . 7 |- (F Fn {A} <-> F:{A}-->ran F)
7	6	biimp 151	. . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8		fndm 3583	. . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
9	8	imaeq2d 3400	. . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10		imadmrn 3410	. . . . . . . . 9 |- (F"dom F) = ran F
11	9, 10	syl5eqr 1519	. . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12		fnsnfv 3762	. . . . . . . . 9 |- ((F Fn {A} /\ A e. {A}) -> {(F` A)} = (F"{A}))
13	2, 12	mpan2 695	. . . . . . . 8 |- (F Fn {A} -> {(F` A)} = (F"{A}))
14	11, 13	eqtr4d 1508	. . . . . . 7 |- (F Fn {A} -> ran F = {(F` A)})
15		feq3 3618	. . . . . . 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 3630	. . . . 5 |- ((F:{A}-->{(F` A)} /\ {(F` A)} (_ B) -> F:{A}-->B)
21	20	ancoms 436	. . . 4 |- (({(F` A)} (_ B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
22		snssi 2463	. . . 4 |- ((F` A) e. B -> {(F` A)} (_ B)
23	21, 22	sylan 448	. . 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 3727	. . . 4 |- (F` A) e. V
26	1, 25	fsn 3829	. . 3 |- (F:{A}-->{(F` A)} <-> F = {<.A, (F` A)>.})
27	26	anbi2i 480	. 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 955   e. wcel 957  Vcvv 1808   (_ wss 2044  {csn 2406  <.cop 2408  dom cdm 3166  ran crn 3167  "cima 3169   Fn wfn 3173  -->wf 3174  ` cfv 3178

This theorem is referenced by:  fnressn 3832  fressnfv 3833  en1 4416

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-reu 1649  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194

Copyright terms: Public domain