Theorem mapsn 6026

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 4111	. . 3 ⊢ {B} ∈ V
3	1, 2	mapval 6011	. 2 ⊢ (A ↑m {B}) = {f ∣ f:{B}–→A}
4		ffn 5223	. . . . . . . 8 ⊢ (f:{B}–→A → f Fn {B})
5		map0.2	. . . . . . . . 9 ⊢ B ∈ V
6	5	snid 3760	. . . . . . . 8 ⊢ B ∈ {B}
7		fneu 5187	. . . . . . . 8 ⊢ ((f Fn {B} ∧ B ∈ {B}) → ∃!y Bfy)
8	4, 6, 7	sylancl 643	. . . . . . 7 ⊢ (f:{B}–→A → ∃!y Bfy)
9		euabsn 3792	. . . . . . . 8 ⊢ (∃!y Bfy ↔ ∃y{y ∣ Bfy} = {y})
10		imadmrn 5008	. . . . . . . . . . . 12 ⊢ (f “ dom f) = ran f
11		fdm 5226	. . . . . . . . . . . . 13 ⊢ (f:{B}–→A → dom f = {B})
12	11	imaeq2d 4942	. . . . . . . . . . . 12 ⊢ (f:{B}–→A → (f “ dom f) = (f “ {B}))
13	10, 12	syl5eqr 2399	. . . . . . . . . . 11 ⊢ (f:{B}–→A → ran f = (f “ {B}))
14		imasn 5018	. . . . . . . . . . 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 5228	. . . . . . . . 9 ⊢ (f:{B}–→A → ran f ⊆ A)
21		sseq1 3292	. . . . . . . . . 10 ⊢ (ran f = {y} → (ran f ⊆ A ↔ {y} ⊆ A))
22		vex 2862	. . . . . . . . . . 11 ⊢ y ∈ V
23	22	snss 3838	. . . . . . . . . 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 5275	. . . . . . . . . . . 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 5269	. . . . . . . . . . 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 5212	. . . . . . . . . 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 5432	. . . . . . . . 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 2620	. . . . 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 5322	. . . . . . . . 9 ⊢ {⟨B, y⟩}:{B}–1-1-onto→{y}
40		f1of 5287	. . . . . . . . 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 5210	. . . . . . . 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 3852	. . . . . . 7 ⊢ (y ∈ A → {y} ⊆ A)
45		fss 5230	. . . . . . 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 2732	. . . 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 2465	. 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 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟨cop 4561   class class class wbr 4639   “ cima 4722  dom cdm 4772  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779  –1-1-onto→wf1o 4780  (class class class)co 5525   ↑_m cmap 5999

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001

This theorem is referenced by: (None)