Theorem mapsn 6902

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 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	. . . 4 ⊢ 𝐴 ∈ V
2		map0.2	. . . . 5 ⊢ 𝐵 ∈ V
3	2	snex 4281	. . . 4 ⊢ {𝐵} ∈ V
4	1, 3	elmap 6889	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5		ffn 5489	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6	2	snid 3704	. . . . . . . 8 ⊢ 𝐵 ∈ {𝐵}
7		fneu 5443	. . . . . . . 8 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
8	5, 6, 7	sylancl 413	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9		euabsn 3745	. . . . . . . 8 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
10		imasng 5108	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
11	2, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}
12		fdm 5495	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
13	12	imaeq2d 5082	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14		imadmrn 5092	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
15	13, 14	eqtr3di 2279	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
16	11, 15	eqtr3id 2278	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
17	16	eqeq1d 2240	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
18	17	exbidv 1873	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
19	9, 18	bitrid 192	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
20	8, 19	mpbid 147	. . . . . 6 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦ran 𝑓 = {𝑦})
21		vex 2806	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	snid 3704	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑦}
23		eleq2 2295	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
24	22, 23	mpbiri 168	. . . . . . . . 9 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25		frn 5498	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3227	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27	24, 26	syl5 32	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦 ∈ 𝐴))
28		dffn4 5574	. . . . . . . . . . . 12 ⊢ (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
29	5, 28	sylib 122	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}–onto→ran 𝑓)
30		fof 5568	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}–onto→ran 𝑓 → 𝑓:{𝐵}⟶ran 𝑓)
31	29, 30	syl 14	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
32		feq3 5474	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
33	31, 32	syl5ibcom 155	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
34	2, 21	fsn 5827	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
35	33, 34	imbitrdi 161	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓 = {⟨𝐵, 𝑦⟩}))
36	27, 35	jcad 307	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
37	36	eximdv 1928	. . . . . 6 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
38	20, 37	mpd 13	. . . . 5 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
39		df-rex 2517	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
40	38, 39	sylibr 134	. . . 4 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
41	2, 21	f1osn 5634	. . . . . . . . 9 ⊢ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42		f1oeq1 5580	. . . . . . . . 9 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
43	41, 42	mpbiri 168	. . . . . . . 8 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}–1-1-onto→{𝑦})
44		f1of 5592	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
45	43, 44	syl 14	. . . . . . 7 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46		snssi 3822	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
47		fss 5501	. . . . . . 7 ⊢ ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
48	45, 46, 47	syl2an 289	. . . . . 6 ⊢ ((𝑓 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 ∈ 𝐴) → 𝑓:{𝐵}⟶𝐴)
49	48	expcom 116	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
50	49	rexlimiv 2645	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)
51	40, 50	impbii 126	. . 3 ⊢ (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
52	4, 51	bitri 184	. 2 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
53	52	abbi2i 2346	1 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541  ∃!weu 2079   ∈ wcel 2202  {cab 2217  ∃wrex 2512  Vcvv 2803   ⊆ wss 3201  {csn 3673  ⟨cop 3676   class class class wbr 4093  dom cdm 4731  ran crn 4732   “ cima 4734   Fn wfn 5328  ⟶wf 5329  –onto→wfo 5331  –1-1-onto→wf1o 5332  (class class class)co 6028   ↑_𝑚 cmap 6860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-map 6862

This theorem is referenced by:  mapsnen  7029