Theorem fsn2 3827

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 2431	. . . . 5 |- A e. {A}
3		ffvelrn 3805	. . . . 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 3619	. . . . 5 |- (F:{A}-->B -> F Fn {A})
6		fnfrn 3625	. . . . . . 7 |- (F Fn {A} <-> F:{A}-->ran F)
7	6	biimp 151	. . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8		fndm 3579	. . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
9	8	imaeq2d 3396	. . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10		imadmrn 3406	. . . . . . . . 9 |- (F"dom F) = ran F
11	9, 10	syl5eqr 1518	. . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12		fnsnfv 3758	. . . . . . . . 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 1507	. . . . . . 7 |- (F Fn {A} -> ran F = {(F` A)})
15		feq3 3614	. . . . . . 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 3626	. . . . 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 2462	. . . 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 3723	. . . 4 |- (F` A) e. V
26	1, 25	fsn 3825	. . 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 954   e. wcel 956  Vcvv 1807   (_ wss 2043  {csn 2405  <.cop 2407  dom cdm 3165  ran crn 3166  "cima 3168   Fn wfn 3172  -->wf 3173  ` cfv 3177

This theorem is referenced by:  fnressn 3828  fressnfv 3829  en1 4413

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-reu 1648  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193

Copyright terms: Public domain