Theorem mapsn 6027

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.)

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 groups:   y,f,A   B,f,y

Proof of Theorem mapsn

Step	Hyp	Ref	Expression
1		map0.1	. . 3 ⊢ A ∈ V
2		snex 4112	. . 3 ⊢ {B} ∈ V
3	1, 2	mapval 6012	. 2 ⊢ (A ↑m {B}) = {f ∣ f:{B}–→A}
4		ffn 5224	. . . . . . . 8 ⊢ (f:{B}–→A → f Fn {B})
5		map0.2	. . . . . . . . 9 ⊢ B ∈ V
6	5	snid 3761	. . . . . . . 8 ⊢ B ∈ {B}
7		fneu 5188	. . . . . . . 8 ⊢ ((f Fn {B} ∧ B ∈ {B}) → ∃!y Bfy)
8	4, 6, 7	sylancl 643	. . . . . . 7 ⊢ (f:{B}–→A → ∃!y Bfy)
9		euabsn 3793	. . . . . . . 8 ⊢ (∃!y Bfy ↔ ∃y{y ∣ Bfy} = {y})
10		imadmrn 5009	. . . . . . . . . . . 12 ⊢ (f “ dom f) = ran f
11		fdm 5227	. . . . . . . . . . . . 13 ⊢ (f:{B}–→A → dom f = {B})
12	11	imaeq2d 4943	. . . . . . . . . . . 12 ⊢ (f:{B}–→A → (f “ dom f) = (f “ {B}))
13	10, 12	syl5eqr 2399	. . . . . . . . . . 11 ⊢ (f:{B}–→A → ran f = (f “ {B}))
14		imasn 5019	. . . . . . . . . . 11 ⊢ (f “ {B}) = {y ∣ Bfy}
15	13, 14	syl6req 2402	. . . . . . . . . 10 ⊢ (f:{B}–→A → {y ∣ Bfy} = ran f)
16	15	eqeq1d 2361	. . . . . . . . 9 ⊢ (f:{B}–→A → ({y ∣ Bfy} = {y} ↔ ran f = {y}))
17	16	exbidv 1626	. . . . . . . 8 ⊢ (f:{B}–→A → (∃y{y ∣ Bfy} = {y} ↔ ∃yran f = {y}))
18	9, 17	syl5bb 248	. . . . . . 7 ⊢ (f:{B}–→A → (∃!y Bfy ↔ ∃yran f = {y}))
19	8, 18	mpbid 201	. . . . . 6 ⊢ (f:{B}–→A → ∃yran f = {y})
20		frn 5229	. . . . . . . . 9 ⊢ (f:{B}–→A → ran f ⊆ A)
21		sseq1 3293	. . . . . . . . . 10 ⊢ (ran f = {y} → (ran f ⊆ A ↔ {y} ⊆ A))
22		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
23	22	snss 3839	. . . . . . . . . 10 ⊢ (y ∈ A ↔ {y} ⊆ A)
24	21, 23	syl6bbr 254	. . . . . . . . 9 ⊢ (ran f = {y} → (ran f ⊆ A ↔ y ∈ A))
25	20, 24	syl5ibcom 211	. . . . . . . 8 ⊢ (f:{B}–→A → (ran f = {y} → y ∈ A))
26		dffn4 5276	. . . . . . . . . . . 12 ⊢ (f Fn {B} ↔ f:{B}–onto→ran f)
27	4, 26	sylib 188	. . . . . . . . . . 11 ⊢ (f:{B}–→A → f:{B}–onto→ran f)
28		fof 5270	. . . . . . . . . . 11 ⊢ (f:{B}–onto→ran f → f:{B}–→ran f)
29	27, 28	syl 15	. . . . . . . . . 10 ⊢ (f:{B}–→A → f:{B}–→ran f)
30		feq3 5213	. . . . . . . . . 10 ⊢ (ran f = {y} → (f:{B}–→ran f ↔ f:{B}–→{y}))
31	29, 30	syl5ibcom 211	. . . . . . . . 9 ⊢ (f:{B}–→A → (ran f = {y} → f:{B}–→{y}))
32	5, 22	fsn 5433	. . . . . . . . 9 ⊢ (f:{B}–→{y} ↔ f = {⟨B, y⟩})
33	31, 32	syl6ib 217	. . . . . . . 8 ⊢ (f:{B}–→A → (ran f = {y} → f = {⟨B, y⟩}))
34	25, 33	jcad 519	. . . . . . 7 ⊢ (f:{B}–→A → (ran f = {y} → (y ∈ A ∧ f = {⟨B, y⟩})))
35	34	eximdv 1622	. . . . . 6 ⊢ (f:{B}–→A → (∃yran f = {y} → ∃y(y ∈ A ∧ f = {⟨B, y⟩})))
36	19, 35	mpd 14	. . . . 5 ⊢ (f:{B}–→A → ∃y(y ∈ A ∧ f = {⟨B, y⟩}))
37		df-rex 2621	. . . . 5 ⊢ (∃y ∈ A f = {⟨B, y⟩} ↔ ∃y(y ∈ A ∧ f = {⟨B, y⟩}))
38	36, 37	sylibr 203	. . . 4 ⊢ (f:{B}–→A → ∃y ∈ A f = {⟨B, y⟩})
39	5, 22	f1osn 5323	. . . . . . . . 9 ⊢ {⟨B, y⟩}:{B}–1-1-onto→{y}
40		f1of 5288	. . . . . . . . 9 ⊢ ({⟨B, y⟩}:{B}–1-1-onto→{y} → {⟨B, y⟩}:{B}–→{y})
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ {⟨B, y⟩}:{B}–→{y}
42		feq1 5211	. . . . . . . 8 ⊢ (f = {⟨B, y⟩} → (f:{B}–→{y} ↔ {⟨B, y⟩}:{B}–→{y}))
43	41, 42	mpbiri 224	. . . . . . 7 ⊢ (f = {⟨B, y⟩} → f:{B}–→{y})
44		snssi 3853	. . . . . . 7 ⊢ (y ∈ A → {y} ⊆ A)
45		fss 5231	. . . . . . 7 ⊢ ((f:{B}–→{y} ∧ {y} ⊆ A) → f:{B}–→A)
46	43, 44, 45	syl2an 463	. . . . . 6 ⊢ ((f = {⟨B, y⟩} ∧ y ∈ A) → f:{B}–→A)
47	46	expcom 424	. . . . 5 ⊢ (y ∈ A → (f = {⟨B, y⟩} → f:{B}–→A))
48	47	rexlimiv 2733	. . . 4 ⊢ (∃y ∈ A f = {⟨B, y⟩} → f:{B}–→A)
49	38, 48	impbii 180	. . 3 ⊢ (f:{B}–→A ↔ ∃y ∈ A f = {⟨B, y⟩})
50	49	abbii 2466	. 2 ⊢ {f ∣ f:{B}–→A} = {f ∣ ∃y ∈ A f = {⟨B, y⟩}}
51	3, 50	eqtri 2373	1 ⊢ (A ↑m {B}) = {f ∣ ∃y ∈ A f = {⟨B, y⟩}}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  {csn 3738  ⟨cop 4562   class class class wbr 4640   “ cima 4723  dom cdm 4773  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780  –1-1-onto→wf1o 4781  (class class class)co 5526   ↑_m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by: (None)