Theorem mapsnd 8820

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 5378	. . . . 5 ⊢ {𝐵} ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝜑 → {𝐵} ∈ V)
4	1, 3	elmapd 8773	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
5		ffn 6659	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6		mapsnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		snidg 4614	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
9		fneu 6599	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
10	5, 8, 9	syl2anr 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
11		euabsn 4680	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
12		frel 6664	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
13		relimasn 6041	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
15		fdm 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
16	15	imaeq2d 6016	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
17		imadmrn 6026	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
18	16, 17	eqtr3di 2783	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
19	14, 18	eqtr3d 2770	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
20	19	eqeq1d 2735	. . . . . . . . . . 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	bitrid 283	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
23	22	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
24	10, 23	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
25		frn 6666	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3929	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27		vsnid 4617	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
28		eleq2 2822	. . . . . . . . . . . . 13 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
29	27, 28	mpbiri 258	. . . . . . . . . . . 12 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
30	26, 29	impel 505	. . . . . . . . . . 11 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
31	30	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ 𝐴)
32		ffrn 6672	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
33		feq3 6639	. . . . . . . . . . . . . 14 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
34	32, 33	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
35	34	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
36	35	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
37	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵 ∈ 𝑊)
38		vex 3441	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
39		fsng 7079	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
42	31, 41	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
43	42	ex 412	. . . . . . . 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 3058	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
47	45, 46	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:{𝐵}⟶𝐴) → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
48	47	ex 412	. . . 4 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
49		f1osng 6813	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
52		f1oeq1 6759	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
53	52	bicomd 223	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
54	53	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
55	51, 54	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
56		f1of 6771	. . . . . . . 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 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
59		snssi 4761	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
60	59	3ad2ant2 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
61	58, 60	fssd 6676	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
62	61	rexlimdv3a 3138	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
63	48, 62	impbid 212	. . 3 ⊢ (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
64	4, 63	bitrd 279	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
65	64	eqabdv 2866	1 ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2565  {cab 2711  ∃wrex 3057  Vcvv 3437   ⊆ wss 3898  {csn 4577  ⟨cop 4583   class class class wbr 5095  dom cdm 5621  ran crn 5622   “ cima 5624  Rel wrel 5626   Fn wfn 6484  ⟶wf 6485  –1-1-onto→wf1o 6488  (class class class)co 7355   ↑_m cmap 8759

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-map 8761

This theorem is referenced by:  mapsn  8822  mapsnend  8969  iunmapsn  45377