Theorem mapsn 3269

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

Hypotheses

Ref	Expression
map0.1	⊢ A ∈ V
map0.2	⊢ B ∈ V

Assertion

Ref	Expression
mapsn	⊢ (A ↑m {B}) = {f∣∃y ∈ A f = {⟨B, y⟩}}

Distinct variable group(s):   y,f,A   B,f,y

Proof of Theorem mapsn

Step	Hyp	Ref	Expression
1		map0.1	. . . 4 ⊢ A ∈ V
2		snex 1859	. . . 4 ⊢ {B} ∈ V
3	1, 2	elmap 3265	. . 3 ⊢ (f ∈ (A ↑m {B}) ↔ f:{B}–→A)
4		map0.2	. . . . . . . . 9 ⊢ B ∈ V
5	4	snid 1830	. . . . . . . 8 ⊢ B ∈ {B}
6		fneu2 2729	. . . . . . . . 9 ⊢ ((f Fn {B} ∧ B ∈ {B}) → ∃!y⟨B, y⟩ ∈ f)
7		ffn 2752	. . . . . . . . 9 ⊢ (f:{B}–→A → f Fn {B})
8	6, 7	sylan 343	. . . . . . . 8 ⊢ ((f:{B}–→A ∧ B ∈ {B}) → ∃!y⟨B, y⟩ ∈ f)
9	5, 8	mpan2 519	. . . . . . 7 ⊢ (f:{B}–→A → ∃!y⟨B, y⟩ ∈ f)
10		frel 2755	. . . . . . . . . . . 12 ⊢ (f:{B}–→A → Rel f)
11		imasn 2616	. . . . . . . . . . . 12 ⊢ (Rel f → (f “ {B}) = {y∣⟨B, y⟩ ∈ f})
12	10, 11	syl 12	. . . . . . . . . . 11 ⊢ (f:{B}–→A → (f “ {B}) = {y∣⟨B, y⟩ ∈ f})
13		fdm 2756	. . . . . . . . . . . . 13 ⊢ (f:{B}–→A → dom f = {B})
14		imaeq2 2603	. . . . . . . . . . . . 13 ⊢ (dom f = {B} → (f “ dom f) = (f “ {B}))
15	13, 14	syl 12	. . . . . . . . . . . 12 ⊢ (f:{B}–→A → (f “ dom f) = (f “ {B}))
16		imadmrn 2610	. . . . . . . . . . . 12 ⊢ (f “ dom f) = ran f
17	15, 16	syl5reqr 1139	. . . . . . . . . . 11 ⊢ (f:{B}–→A → (f “ {B}) = ran f)
18	12, 17	eqtr3d 1130	. . . . . . . . . 10 ⊢ (f:{B}–→A → {y∣⟨B, y⟩ ∈ f} = ran f)
19	18	cleq1d 1109	. . . . . . . . 9 ⊢ (f:{B}–→A → ({y∣⟨B, y⟩ ∈ f} = {y} ↔ ran f = {y}))
20	19	biexdv 936	. . . . . . . 8 ⊢ (f:{B}–→A → (∃y{y∣⟨B, y⟩ ∈ f} = {y} ↔ ∃yran f = {y}))
21		eusn 1913	. . . . . . . 8 ⊢ (∃!y⟨B, y⟩ ∈ f ↔ ∃y{y∣⟨B, y⟩ ∈ f} = {y})
22	20, 21	syl5bb 410	. . . . . . 7 ⊢ (f:{B}–→A → (∃!y⟨B, y⟩ ∈ f ↔ ∃yran f = {y}))
23	9, 22	mpbid 170	. . . . . 6 ⊢ (f:{B}–→A → ∃yran f = {y})
24		frn 2757	. . . . . . . . . 10 ⊢ (f:{B}–→A → ran f ⊆ A)
25	24	sseld 1506	. . . . . . . . 9 ⊢ (f:{B}–→A → (y ∈ ran f → y ∈ A))
26		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
27	26	snid 1830	. . . . . . . . . 10 ⊢ y ∈ {y}
28		eleq2 1150	. . . . . . . . . 10 ⊢ (ran f = {y} → (y ∈ ran f ↔ y ∈ {y}))
29	27, 28	mpbiri 169	. . . . . . . . 9 ⊢ (ran f = {y} → y ∈ ran f)
30	25, 29	syl5 22	. . . . . . . 8 ⊢ (f:{B}–→A → (ran f = {y} → y ∈ A))
31		feq3 2750	. . . . . . . . . . 11 ⊢ (ran f = {y} → (f:{B}–→ran f ↔ f:{B}–→{y}))
32		fnforn 2791	. . . . . . . . . . . . 13 ⊢ (f Fn {B} ↔ f:{B}–onto→ran f)
33	7, 32	sylib 173	. . . . . . . . . . . 12 ⊢ (f:{B}–→A → f:{B}–onto→ran f)
34		fof 2788	. . . . . . . . . . . 12 ⊢ (f:{B}–onto→ran f → f:{B}–→ran f)
35	33, 34	syl 12	. . . . . . . . . . 11 ⊢ (f:{B}–→A → f:{B}–→ran f)
36	31, 35	syl5bi 183	. . . . . . . . . 10 ⊢ (ran f = {y} → (f:{B}–→A → f:{B}–→{y}))
37	36	com12 13	. . . . . . . . 9 ⊢ (f:{B}–→A → (ran f = {y} → f:{B}–→{y}))
38	4, 26	fsn 2895	. . . . . . . . 9 ⊢ (f:{B}–→{y} ↔ f = {⟨B, y⟩})
39	37, 38	syl6ib 185	. . . . . . . 8 ⊢ (f:{B}–→A → (ran f = {y} → f = {⟨B, y⟩}))
40	30, 39	jcad 455	. . . . . . 7 ⊢ (f:{B}–→A → (ran f = {y} → (y ∈ A ∧ f = {⟨B, y⟩})))
41	40	19.22dv 947	. . . . . 6 ⊢ (f:{B}–→A → (∃yran f = {y} → ∃y(y ∈ A ∧ f = {⟨B, y⟩})))
42	23, 41	mpd 46	. . . . 5 ⊢ (f:{B}–→A → ∃y(y ∈ A ∧ f = {⟨B, y⟩}))
43		df-rex 1206	. . . . 5 ⊢ (∃y ∈ A f = {⟨B, y⟩} ↔ ∃y(y ∈ A ∧ f = {⟨B, y⟩}))
44	42, 43	sylibr 175	. . . 4 ⊢ (f:{B}–→A → ∃y ∈ A f = {⟨B, y⟩})
45		fss 2759	. . . . . . . 8 ⊢ ((f:{B}–→{y} ∧ {y} ⊆ A) → f:{B}–→A)
46	4, 26	f1osn 2827	. . . . . . . . . 10 ⊢ {⟨B, y⟩}:{B}–1-1-onto→{y}
47		f1oeq1 2795	. . . . . . . . . 10 ⊢ (f = {⟨B, y⟩} → (f:{B}–1-1-onto→{y} ↔ {⟨B, y⟩}:{B}–1-1-onto→{y}))
48	46, 47	mpbiri 169	. . . . . . . . 9 ⊢ (f = {⟨B, y⟩} → f:{B}–1-1-onto→{y})
49		f1of 2800	. . . . . . . . 9 ⊢ (f:{B}–1-1-onto→{y} → f:{B}–→{y})
50	48, 49	syl 12	. . . . . . . 8 ⊢ (f = {⟨B, y⟩} → f:{B}–→{y})
51		snssi 1851	. . . . . . . 8 ⊢ (y ∈ A → {y} ⊆ A)
52	45, 50, 51	syl2an 349	. . . . . . 7 ⊢ ((f = {⟨B, y⟩} ∧ y ∈ A) → f:{B}–→A)
53	52	exp 291	. . . . . 6 ⊢ (f = {⟨B, y⟩} → (y ∈ A → f:{B}–→A))
54	53	com12 13	. . . . 5 ⊢ (y ∈ A → (f = {⟨B, y⟩} → f:{B}–→A))
55	54	r19.23aiv 1284	. . . 4 ⊢ (∃y ∈ A f = {⟨B, y⟩} → f:{B}–→A)
56	44, 55	impbi 139	. . 3 ⊢ (f:{B}–→A ↔ ∃y ∈ A f = {⟨B, y⟩})
57	3, 56	bitr 151	. 2 ⊢ (f ∈ (A ↑m {B}) ↔ ∃y ∈ A f = {⟨B, y⟩})
58	57	biabri 1180	1 ⊢ (A ↑m {B}) = {f∣∃y ∈ A f = {⟨B, y⟩}}

Syntax hints:   ∧ wa 196  ∃wex 678  ∃!weu 1007  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  {csn 1808  ⟨cop 1810  dom cdm 2410  ran crn 2411   “ cima 2413  Rel wrel 2415   Fn wfn 2417  –→wf 2418  –onto→wfo 2420  –1-1-onto→wf1o 2421  (class class class)co 3001   ↑_m cm 3258

This theorem is referenced by:  mapsnen 3334

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-opr 3003  df-oprab 3004  df-map 3259

metamath.org