Theorem mapsnd 39388

Description: The value of set exponentiation with a singleton exponent. (Contributed 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		mapsnd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		snex 4908	. . . . . 6 ⊢ {𝐵} ∈ V
4	3	a1i 11	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → {𝐵} ∈ V)
6		elmapg 7870	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐵} ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
7	1, 5, 6	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
8		ffn 6045	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
9	8	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵}))
10	9	imp 445	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → 𝑓 Fn {𝐵})
11		snidg 4206	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
12	2, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → 𝐵 ∈ {𝐵})
14		fneu 5995	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
15	10, 13, 14	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
16		euabsn 4261	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
17		frel 6050	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
18		relimasn 5488	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
19	17, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
20		imadmrn 5476	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
21		fdm 6051	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
22	21	imaeq2d 5466	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
23	20, 22	syl5reqr 2671	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
24	19, 23	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
25	24	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
26	25	exbidv 1850	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
27	16, 26	syl5bb 272	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
28	27	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
29	15, 28	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
30		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
31	30	snid 4208	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ {𝑦}
32		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
33	31, 32	mpbiri 248	. . . . . . . . . . . . 13 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
34		frn 6053	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
35	34	sseld 3602	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
36	33, 35	syl5 34	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦 ∈ 𝐴))
37	36	imp 445	. . . . . . . . . . 11 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
38	37	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
39		dffn4 6121	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
40	8, 39	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}–onto→ran 𝑓)
41		fof 6115	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}–onto→ran 𝑓 → 𝑓:{𝐵}⟶ran 𝑓)
42	40, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
43		feq3 6028	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
44	42, 43	syl5ibcom 235	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
45	44	imp 445	. . . . . . . . . . . 12 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
46	45	adantll 750	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
47	2	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊)
48	30	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ V)
49		fsng 6404	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
50	47, 48, 49	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
51	46, 50	mpbid 222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
52	38, 51	jca 554	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
53	52	ex 450	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
54	53	eximdv 1846	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
55	29, 54	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
56		df-rex 2918	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
57	55, 56	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
58	57	ex 450	. . . 4 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
59	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑦 ∈ V)
60		f1osng 6177	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
61	2, 59, 60	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
62	61	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
63		f1oeq1 6127	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
64	63	bicomd 213	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
65	64	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
66	62, 65	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
67		f1of 6137	. . . . . . . . 9 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
68	66, 67	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
69	68	3adant2 1080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
70		snssi 4339	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
71	70	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
72		fss 6056	. . . . . . 7 ⊢ ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
73	69, 71, 72	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
74	73	3exp 1264	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)))
75	74	rexlimdv 3030	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
76	58, 75	impbid 202	. . 3 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
77	7, 76	bitrd 268	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
78	77	abbi2dv 2742	1 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  {cab 2608  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  {csn 4177  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  ran crn 5115   “ cima 5117  Rel wrel 5119   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  mapsnend  39391  iunmapsn  39409