Theorem mapsnd 8674

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
mapsnd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapsnd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

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

Distinct variable groups:   𝐴,𝑓,𝑦   𝐵,𝑓,𝑦   𝜑,𝑓,𝑦

Allowed substitution hints:   𝑉(𝑦,𝑓)   𝑊(𝑦,𝑓)

Proof of Theorem mapsnd

Step	Hyp	Ref	Expression
1		mapsnd.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		snex 5354	. . . . 5 ⊢ {𝐵} ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → {𝐵} ∈ V)
4	1, 3	elmapd 8629	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
5		ffn 6600	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6		mapsnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		snidg 4595	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
9		fneu 6543	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
10	5, 8, 9	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
11		euabsn 4662	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
12		frel 6605	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
13		relimasn 5992	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
15		fdm 6609	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
16	15	imaeq2d 5969	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
17		imadmrn 5979	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
18	16, 17	eqtr3di 2793	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
19	14, 18	eqtr3d 2780	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
20	19	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
21	20	exbidv 1924	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
22	11, 21	bitrid 282	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
23	22	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
24	10, 23	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
25		frn 6607	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3920	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27		vsnid 4598	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
28		eleq2 2827	. . . . . . . . . . . . 13 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
29	27, 28	mpbiri 257	. . . . . . . . . . . 12 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
30	26, 29	impel 506	. . . . . . . . . . 11 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
31	30	adantll 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
32		ffrn 6614	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
33		feq3 6583	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
34	32, 33	syl5ibcom 244	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
35	34	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
36	35	adantll 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
37	6	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊)
38		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
39		fsng 7009	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
40	37, 38, 39	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
41	36, 40	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
42	31, 41	jca 512	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
43	42	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
44	43	eximdv 1920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
45	24, 44	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
46		df-rex 3070	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
47	45, 46	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
48	47	ex 413	. . . 4 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
49		f1osng 6757	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
50	6, 38, 49	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
51	50	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
52		f1oeq1 6704	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
53	52	bicomd 222	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
54	53	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
55	51, 54	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
56		f1of 6716	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
57	55, 56	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
58	57	3adant2 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
59		snssi 4741	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
60	59	3ad2ant2 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
61	58, 60	fssd 6618	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
62	61	rexlimdv3a 3215	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
63	48, 62	impbid 211	. . 3 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
64	4, 63	bitrd 278	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
65	64	abbi2dv 2877	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃!weu 2568  {cab 2715  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  {csn 4561  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  ran crn 5590   “ cima 5592  Rel wrel 5594   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617

This theorem is referenced by:  mapsn  8676  mapsnend  8826  iunmapsn  42757