Theorem mapsnd 8886

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 5412	. . . . 5 ⊢ {𝐵} ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → {𝐵} ∈ V)
4	1, 3	elmapd 8839	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
5		ffn 6709	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6		mapsnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		snidg 4628	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
8	6, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
9		fneu 6649	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
10	5, 8, 9	syl2anr 609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
11		euabsn 4694	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
12		frel 6715	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
13		relimasn 6089	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
14	12, 13	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
15		fdm 6719	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
16	15	imaeq2d 6064	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
17		imadmrn 6074	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
18	16, 17	eqtr3di 2815	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
19	14, 18	eqtr3d 2802	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
20	19	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
21	20	exbidv 1954	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
22	11, 21	bitrid 286	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
23	22	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
24	10, 23	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
25		frn 6717	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3937	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27		vsnid 4631	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
28		eleq2 2854	. . . . . . . . . . . . 13 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
29	27, 28	mpbiri 261	. . . . . . . . . . . 12 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
30	26, 29	impel 515	. . . . . . . . . . 11 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
31	30	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
32		ffrn 6723	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
33		feq3 6689	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
34	32, 33	syl5ibcom 248	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
35	34	imp 412	. . . . . . . . . . . 12 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
36	35	adantll 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
37	6	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊)
38		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
39		fsng 7137	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
40	37, 38, 39	sylancl 598	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
41	36, 40	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
42	31, 41	jca 521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
43	42	ex 418	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
44	43	eximdv 1950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
45	24, 44	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
46		df-rex 3092	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
47	45, 46	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
48	47	ex 418	. . . 4 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
49		f1osng 6867	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
50	6, 38, 49	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
51	50	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
52		f1oeq1 6812	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
53	52	bicomd 226	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
54	53	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
55	51, 54	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
56		f1of 6824	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
57	55, 56	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
58	57	3adant2 1149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
59		snssi 4753	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
60	59	3ad2ant2 1152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
61	58, 60	fssd 6727	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
62	61	rexlimdv3a 3172	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
63	48, 62	impbid 215	. . 3 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
64	4, 63	bitrd 282	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
65	64	eqabdv 2898	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃!weu 2598  {cab 2743  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  {csn 4591  ⟨cop 4597   class class class wbr 5111  dom cdm 5663  ran crn 5664   “ cima 5666  Rel wrel 5668   Fn wfn 6535  ⟶wf 6536  –1-1-onto→wf1o 6539  (class class class)co 7416   ↑_m cmap 8826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8828

This theorem is used by:  mapsn  8888  mapsnend  9036  iunmapsn  45966