Theorem mapsn 5404

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89.

Hypotheses

Ref	Expression
map0.1	|- A e. _V
map0.2	|- B e. _V

Assertion

Ref	Expression
mapsn	|- (A ^m {B}) = {f | E.y e. A f = {<.B, y>.}}

Distinct variable groups:   y,f,A   B,f,y

Proof of Theorem mapsn

Step	Hyp	Ref	Expression
1		map0.1	. . . 4 |- A e. _V
2		snex 3492	. . . 4 |- {B} e. _V
3	1, 2	elmap 5393	. . 3 |- (f e. (A ^m {B}) <-> f:{B}-->A)
4		fneu 4517	. . . . . . . 8 |- ((f Fn {B} /\ B e. {B}) -> E!y Bfy)
5		ffn 4562	. . . . . . . 8 |- (f:{B}-->A -> f Fn {B})
6		map0.2	. . . . . . . . 9 |- B e. _V
7	6	snid 3069	. . . . . . . 8 |- B e. {B}
8	4, 5, 7	sylancl 525	. . . . . . 7 |- (f:{B}-->A -> E!y Bfy)
9		frel 4566	. . . . . . . . . . . 12 |- (f:{B}-->A -> Rel f)
10		relimasn 4288	. . . . . . . . . . . 12 |- (Rel f -> (f"{B}) = {y | Bfy})
11	9, 10	syl 12	. . . . . . . . . . 11 |- (f:{B}-->A -> (f"{B}) = {y | Bfy})
12		fdm 4567	. . . . . . . . . . . . 13 |- (f:{B}-->A -> dom f = {B})
13	12	imaeq2d 4264	. . . . . . . . . . . 12 |- (f:{B}-->A -> (f"dom f) = (f"{B}))
14		imadmrn 4277	. . . . . . . . . . . 12 |- (f"dom f) = ran f
15	13, 14	syl5reqr 1943	. . . . . . . . . . 11 |- (f:{B}-->A -> (f"{B}) = ran f)
16	11, 15	eqtr3d 1927	. . . . . . . . . 10 |- (f:{B}-->A -> {y | Bfy} = ran f)
17	16	eqeq1d 1892	. . . . . . . . 9 |- (f:{B}-->A -> ({y | Bfy} = {y} <-> ran f = {y}))
18	17	exbidv 1657	. . . . . . . 8 |- (f:{B}-->A -> (E.y{y | Bfy} = {y} <-> E.yran f = {y}))
19		euabsn 3095	. . . . . . . 8 |- (E!y Bfy <-> E.y{y | Bfy} = {y})
20	18, 19	syl5bb 591	. . . . . . 7 |- (f:{B}-->A -> (E!y Bfy <-> E.yran f = {y}))
21	8, 20	mpbid 212	. . . . . 6 |- (f:{B}-->A -> E.yran f = {y})
22		frn 4569	. . . . . . . . . 10 |- (f:{B}-->A -> ran f C_ A)
23	22	sseld 2619	. . . . . . . . 9 |- (f:{B}-->A -> (y e. ran f -> y e. A))
24		visset 2295	. . . . . . . . . . 11 |- y e. _V
25	24	snid 3069	. . . . . . . . . 10 |- y e. {y}
26		eleq2 1958	. . . . . . . . . 10 |- (ran f = {y} -> (y e. ran f <-> y e. {y}))
27	25, 26	mpbiri 211	. . . . . . . . 9 |- (ran f = {y} -> y e. ran f)
28	23, 27	syl5 20	. . . . . . . 8 |- (f:{B}-->A -> (ran f = {y} -> y e. A))
29		feq3 4553	. . . . . . . . . 10 |- (ran f = {y} -> (f:{B}-->ran f <-> f:{B}-->{y}))
30		dffn4 4623	. . . . . . . . . . . 12 |- (f Fn {B} <-> f:{B}-onto->ran f)
31	5, 30	sylib 215	. . . . . . . . . . 11 |- (f:{B}-->A -> f:{B}-onto->ran f)
32		fof 4617	. . . . . . . . . . 11 |- (f:{B}-onto->ran f -> f:{B}-->ran f)
33	31, 32	syl 12	. . . . . . . . . 10 |- (f:{B}-->A -> f:{B}-->ran f)
34	29, 33	syl5cbi 226	. . . . . . . . 9 |- (f:{B}-->A -> (ran f = {y} -> f:{B}-->{y}))
35	6, 24	fsn 4807	. . . . . . . . 9 |- (f:{B}-->{y} <-> f = {<.B, y>.})
36	34, 35	syl6ib 229	. . . . . . . 8 |- (f:{B}-->A -> (ran f = {y} -> f = {<.B, y>.}))
37	28, 36	jcad 661	. . . . . . 7 |- (f:{B}-->A -> (ran f = {y} -> (y e. A /\ f = {<.B, y>.})))
38	37	eximdv 1669	. . . . . 6 |- (f:{B}-->A -> (E.yran f = {y} -> E.y(y e. A /\ f = {<.B, y>.})))
39	21, 38	mpd 29	. . . . 5 |- (f:{B}-->A -> E.y(y e. A /\ f = {<.B, y>.}))
40		df-rex 2110	. . . . 5 |- (E.y e. A f = {<.B, y>.} <-> E.y(y e. A /\ f = {<.B, y>.}))
41	39, 40	sylibr 217	. . . 4 |- (f:{B}-->A -> E.y e. A f = {<.B, y>.})
42		fss 4571	. . . . . . 7 |- ((f:{B}-->{y} /\ {y} C_ A) -> f:{B}-->A)
43	6, 24	f1osn 4674	. . . . . . . . 9 |- {<.B, y>.}:{B}-1-1-onto->{y}
44		f1oeq1 4630	. . . . . . . . 9 |- (f = {<.B, y>.} -> (f:{B}-1-1-onto->{y} <-> {<.B, y>.}:{B}-1-1-onto->{y}))
45	43, 44	mpbiri 211	. . . . . . . 8 |- (f = {<.B, y>.} -> f:{B}-1-1-onto->{y})
46		f1of 4635	. . . . . . . 8 |- (f:{B}-1-1-onto->{y} -> f:{B}-->{y})
47	45, 46	syl 12	. . . . . . 7 |- (f = {<.B, y>.} -> f:{B}-->{y})
48		snssi 3129	. . . . . . 7 |- (y e. A -> {y} C_ A)
49	42, 47, 48	syl2an 503	. . . . . 6 |- ((f = {<.B, y>.} /\ y e. A) -> f:{B}-->A)
50	49	expcom 403	. . . . 5 |- (y e. A -> (f = {<.B, y>.} -> f:{B}-->A))
51	50	r19.23aiv 2211	. . . 4 |- (E.y e. A f = {<.B, y>.} -> f:{B}-->A)
52	41, 51	impbii 174	. . 3 |- (f:{B}-->A <-> E.y e. A f = {<.B, y>.})
53	3, 52	bitri 190	. 2 |- (f e. (A ^m {B}) <-> E.y e. A f = {<.B, y>.})
54	53	abbi2i 2005	1 |- (A ^m {B}) = {f | E.y e. A f = {<.B, y>.}}

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  E.wrex 2106  _Vcvv 2292   C_ wss 2593  {csn 3044  <.cop 3046   class class class wbr 3338  dom cdm 3986  ran crn 3987  "cima 3989  Rel wrel 3991   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  -1-1-onto->wf1o 3997  (class class class)co 4884   ^m cmap 5381

This theorem is referenced by:  mapsnen 5488  ismrer1 16024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383

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