Theorem mapsnd 8452

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 5334	. . . . 5 ⊢ {𝐵} ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → {𝐵} ∈ V)
4	1, 3	elmapd 8422	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
5		ffn 6516	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6		mapsnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		snidg 4601	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
9		fneu 6463	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
10	5, 8, 9	syl2anr 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
11		euabsn 4664	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
12		frel 6521	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
13		relimasn 5954	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
15		imadmrn 5941	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
16		fdm 6524	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
17	16	imaeq2d 5931	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
18	15, 17	syl5reqr 2873	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
19	14, 18	eqtr3d 2860	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
20	19	eqeq1d 2825	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
21	20	exbidv 1922	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
22	11, 21	syl5bb 285	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
23	22	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
24	10, 23	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
25		frn 6522	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3968	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27		vsnid 4604	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
28		eleq2 2903	. . . . . . . . . . . . 13 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
29	27, 28	mpbiri 260	. . . . . . . . . . . 12 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
30	26, 29	impel 508	. . . . . . . . . . 11 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
31	30	adantll 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
32		ffrn 6528	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
33		feq3 6499	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
34	32, 33	syl5ibcom 247	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
35	34	imp 409	. . . . . . . . . . . 12 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
36	35	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
37	6	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊)
38		vex 3499	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
39		fsng 6901	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
40	37, 38, 39	sylancl 588	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
41	36, 40	mpbid 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
42	31, 41	jca 514	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
43	42	ex 415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
44	43	eximdv 1918	. . . . . . 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 3146	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
47	45, 46	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
48	47	ex 415	. . . 4 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
49		f1osng 6657	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
50	6, 38, 49	sylancl 588	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
51	50	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
52		f1oeq1 6606	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
53	52	bicomd 225	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
54	53	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
55	51, 54	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
56		f1of 6617	. . . . . . . 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 1127	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
59		snssi 4743	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
60	59	3ad2ant2 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
61	58, 60	fssd 6530	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
62	61	rexlimdv3a 3288	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
63	48, 62	impbid 214	. . 3 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
64	4, 63	bitrd 281	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
65	64	abbi2dv 2952	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  {cab 2801  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  {csn 4569  ⟨cop 4575   class class class wbr 5068  dom cdm 5557  ran crn 5558   “ cima 5560  Rel wrel 5562   Fn wfn 6352  ⟶wf 6353  –1-1-onto→wf1o 6356  (class class class)co 7158   ↑_m cmap 8408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410

This theorem is referenced by:  mapsn  8454  mapsnend  8590  iunmapsn  41487