Theorem mapsn 4329

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 2740	. . . 4 |- {B} e. V
3	1, 2	elmap 4318	. . 3 |- (f e. (A ^m {B}) <-> f:{B}-->A)
4		map0.2	. . . . . . . . 9 |- B e. V
5	4	snid 2425	. . . . . . . 8 |- B e. {B}
6		fneu 3578	. . . . . . . . 9 |- ((f Fn {B} /\ B e. {B}) -> E!y Bfy)
7		ffn 3613	. . . . . . . . 9 |- (f:{B}-->A -> f Fn {B})
8	6, 7	sylan 448	. . . . . . . 8 |- ((f:{B}-->A /\ B e. {B}) -> E!y Bfy)
9	5, 8	mpan2 694	. . . . . . 7 |- (f:{B}-->A -> E!y Bfy)
10		frel 3616	. . . . . . . . . . . 12 |- (f:{B}-->A -> Rel f)
11		relimasn 3409	. . . . . . . . . . . 12 |- (Rel f -> (f"{B}) = {y | Bfy})
12	10, 11	syl 10	. . . . . . . . . . 11 |- (f:{B}-->A -> (f"{B}) = {y | Bfy})
13		fdm 3617	. . . . . . . . . . . . 13 |- (f:{B}-->A -> dom f = {B})
14	13	imaeq2d 3388	. . . . . . . . . . . 12 |- (f:{B}-->A -> (f"dom f) = (f"{B}))
15		imadmrn 3398	. . . . . . . . . . . 12 |- (f"dom f) = ran f
16	14, 15	syl5reqr 1514	. . . . . . . . . . 11 |- (f:{B}-->A -> (f"{B}) = ran f)
17	12, 16	eqtr3d 1501	. . . . . . . . . 10 |- (f:{B}-->A -> {y | Bfy} = ran f)
18	17	eqeq1d 1475	. . . . . . . . 9 |- (f:{B}-->A -> ({y | Bfy} = {y} <-> ran f = {y}))
19	18	exbidv 1274	. . . . . . . 8 |- (f:{B}-->A -> (E.y{y | Bfy} = {y} <-> E.yran f = {y}))
20		eusn 2436	. . . . . . . 8 |- (E!y Bfy <-> E.y{y | Bfy} = {y})
21	19, 20	syl5bb 530	. . . . . . 7 |- (f:{B}-->A -> (E!y Bfy <-> E.yran f = {y}))
22	9, 21	mpbid 195	. . . . . 6 |- (f:{B}-->A -> E.yran f = {y})
23		frn 3618	. . . . . . . . . 10 |- (f:{B}-->A -> ran f (_ A)
24	23	sseld 2057	. . . . . . . . 9 |- (f:{B}-->A -> (y e. ran f -> y e. A))
25		visset 1804	. . . . . . . . . . 11 |- y e. V
26	25	snid 2425	. . . . . . . . . 10 |- y e. {y}
27		eleq2 1527	. . . . . . . . . 10 |- (ran f = {y} -> (y e. ran f <-> y e. {y}))
28	26, 27	mpbiri 194	. . . . . . . . 9 |- (ran f = {y} -> y e. ran f)
29	24, 28	syl5 21	. . . . . . . 8 |- (f:{B}-->A -> (ran f = {y} -> y e. A))
30		feq3 3608	. . . . . . . . . 10 |- (ran f = {y} -> (f:{B}-->ran f <-> f:{B}-->{y}))
31		fnforn 3662	. . . . . . . . . . . 12 |- (f Fn {B} <-> f:{B}-onto->ran f)
32	7, 31	sylib 198	. . . . . . . . . . 11 |- (f:{B}-->A -> f:{B}-onto->ran f)
33		fof 3657	. . . . . . . . . . 11 |- (f:{B}-onto->ran f -> f:{B}-->ran f)
34	32, 33	syl 10	. . . . . . . . . 10 |- (f:{B}-->A -> f:{B}-->ran f)
35	30, 34	syl5cbi 209	. . . . . . . . 9 |- (f:{B}-->A -> (ran f = {y} -> f:{B}-->{y}))
36	4, 25	fsn 3819	. . . . . . . . 9 |- (f:{B}-->{y} <-> f = {<.B, y>.})
37	35, 36	syl6ib 212	. . . . . . . 8 |- (f:{B}-->A -> (ran f = {y} -> f = {<.B, y>.}))
38	29, 37	jcad 598	. . . . . . 7 |- (f:{B}-->A -> (ran f = {y} -> (y e. A /\ f = {<.B, y>.})))
39	38	19.22dv 1285	. . . . . 6 |- (f:{B}-->A -> (E.yran f = {y} -> E.y(y e. A /\ f = {<.B, y>.})))
40	22, 39	mpd 26	. . . . 5 |- (f:{B}-->A -> E.y(y e. A /\ f = {<.B, y>.}))
41		df-rex 1642	. . . . 5 |- (E.y e. A f = {<.B, y>.} <-> E.y(y e. A /\ f = {<.B, y>.}))
42	40, 41	sylibr 200	. . . 4 |- (f:{B}-->A -> E.y e. A f = {<.B, y>.})
43		fss 3620	. . . . . . 7 |- ((f:{B}-->{y} /\ {y} (_ A) -> f:{B}-->A)
44	4, 25	f1osn 3704	. . . . . . . . 9 |- {<.B, y>.}:{B}-1-1-onto->{y}
45		f1oeq1 3669	. . . . . . . . 9 |- (f = {<.B, y>.} -> (f:{B}-1-1-onto->{y} <-> {<.B, y>.}:{B}-1-1-onto->{y}))
46	44, 45	mpbiri 194	. . . . . . . 8 |- (f = {<.B, y>.} -> f:{B}-1-1-onto->{y})
47		f1of 3674	. . . . . . . 8 |- (f:{B}-1-1-onto->{y} -> f:{B}-->{y})
48	46, 47	syl 10	. . . . . . 7 |- (f = {<.B, y>.} -> f:{B}-->{y})
49		snssi 2457	. . . . . . 7 |- (y e. A -> {y} (_ A)
50	43, 48, 49	syl2an 454	. . . . . 6 |- ((f = {<.B, y>.} /\ y e. A) -> f:{B}-->A)
51	50	expcom 374	. . . . 5 |- (y e. A -> (f = {<.B, y>.} -> f:{B}-->A))
52	51	r19.23aiv 1735	. . . 4 |- (E.y e. A f = {<.B, y>.} -> f:{B}-->A)
53	42, 52	impbi 157	. . 3 |- (f:{B}-->A <-> E.y e. A f = {<.B, y>.})
54	3, 53	bitr 173	. 2 |- (f e. (A ^m {B}) <-> E.y e. A f = {<.B, y>.})
55	54	abbi2i 1566	1 |- (A ^m {B}) = {f | E.y e. A f = {<.B, y>.}}

Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  {cab 1456  E.wrex 1638  Vcvv 1802   (_ wss 2037  {csn 2399  <.cop 2401   class class class wbr 2609  dom cdm 3160  ran crn 3161  "cima 3163  Rel wrel 3165   Fn wfn 3167  -->wf 3168  -onto->wfo 3170  -1-1-onto->wf1o 3171  (class class class)co 3948   ^m cm 4306

This theorem is referenced by:  mapsnen 4410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-reu 1643  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308

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