Theorem mapsn 6715

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 4203	. . . 4 ⊢ {𝐵} ∈ V
4	1, 3	elmap 6702	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5		ffn 5384	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6	2	snid 3638	. . . . . . . 8 ⊢ 𝐵 ∈ {𝐵}
7		fneu 5339	. . . . . . . 8 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
8	5, 6, 7	sylancl 413	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9		euabsn 3677	. . . . . . . 8 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
10		imasng 5011	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
11	2, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}
12		fdm 5390	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
13	12	imaeq2d 4988	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14		imadmrn 4998	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
15	13, 14	eqtr3di 2237	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
16	11, 15	eqtr3id 2236	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
17	16	eqeq1d 2198	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
18	17	exbidv 1836	. . . . . . . 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 2755	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	snid 3638	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑦}
23		eleq2 2253	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
24	22, 23	mpbiri 168	. . . . . . . . 9 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25		frn 5393	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3169	. . . . . . . . 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 5463	. . . . . . . . . . . 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 5457	. . . . . . . . . . 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 5369	. . . . . . . . . 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 5708	. . . . . . . . 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 1891	. . . . . 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 2474	. . . . 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 5520	. . . . . . . . 9 ⊢ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42		f1oeq1 5468	. . . . . . . . 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 5480	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
45	43, 44	syl 14	. . . . . . 7 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46		snssi 3751	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
47		fss 5396	. . . . . . 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 2601	. . . 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 2304	1 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503  ∃!weu 2038   ∈ wcel 2160  {cab 2175  ∃wrex 2469  Vcvv 2752   ⊆ wss 3144  {csn 3607  ⟨cop 3610   class class class wbr 4018  dom cdm 4644  ran crn 4645   “ cima 4647   Fn wfn 5230  ⟶wf 5231  –onto→wfo 5233  –1-1-onto→wf1o 5234  (class class class)co 5895   ↑_𝑚 cmap 6673

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-map 6675

This theorem is referenced by:  mapsnen  6836