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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
StepHypRef Expression
1 fsn2.1 . . . . . 6 |- A e. V
21snid 2480 . . . . 5 |- A e. {A}
3 ffvelrn 3890 . . . . 5 |- ((F:{A}-->B /\ A e. {A}) -> (F` A) e. B)
42, 3mpan2 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)
76biimpi 149 . . . . . 6 |- (F Fn {A} -> F:{A}-->ran F)
8 fndm 3662 . . . . . . . . . 10 |- (F Fn {A} -> dom F = {A})
98imaeq2d 3467 . . . . . . . . 9 |- (F Fn {A} -> (F"dom F) = (F"{A}))
10 imadmrn 3477 . . . . . . . . 9 |- (F"dom F) = ran F
119, 10syl5eqr 1558 . . . . . . . 8 |- (F Fn {A} -> ran F = (F"{A}))
12 fnsnfv 3843 . . . . . . . . 9 |- ((F Fn {A} /\ A e. {A}) -> {(F` A)} = (F"{A}))
132, 12mpan2 699 . . . . . . . 8 |- (F Fn {A} -> {(F` A)} = (F"{A}))
1411, 13eqtr4d 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)}))
1614, 15syl 10 . . . . . 6 |- (F Fn {A} -> (F:{A}-->ran F <-> F:{A}-->{(F` A)}))
177, 16mpbid 193 . . . . 5 |- (F Fn {A} -> F:{A}-->{(F` A)})
185, 17syl 10 . . . 4 |- (F:{A}-->B -> F:{A}-->{(F` A)})
194, 18jca 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)
2120ancoms 438 . . . 4 |- (({(F` A)} (_ B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
22 snssi 2514 . . . 4 |- ((F` A) e. B -> {(F` A)} (_ B)
2321, 22sylan 450 . . 3 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) -> F:{A}-->B)
2419, 23impbii 155 . 2 |- (F:{A}-->B <-> ((F` A) e. B /\ F:{A}-->{(F` A)}))
25 fvex 3808 . . . 4 |- (F` A) e. V
261, 25fsn 3910 . . 3 |- (F:{A}-->{(F` A)} <-> F = {<.A, (F` A)>.})
2726anbi2i 482 . 2 |- (((F` A) e. B /\ F:{A}-->{(F` A)}) <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
2824, 27bitri 171 1 |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
Colors of variables: wff set class
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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