Theorem mapsn 6684

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

Hypotheses

Ref	Expression
map0.1	⊢ 𝐴 ∈ V
map0.2	⊢ 𝐵 ∈ V

Assertion

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

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

Proof of Theorem mapsn

Step	Hyp	Ref	Expression
1		map0.1	. . . 4 ⊢ 𝐴 ∈ V
2		map0.2	. . . . 5 ⊢ 𝐵 ∈ V
3	2	snex 4182	. . . 4 ⊢ {𝐵} ∈ V
4	1, 3	elmap 6671	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5		ffn 5361	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6	2	snid 3622	. . . . . . . 8 ⊢ 𝐵 ∈ {𝐵}
7		fneu 5316	. . . . . . . 8 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
8	5, 6, 7	sylancl 413	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9		euabsn 3661	. . . . . . . 8 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
10		imasng 4989	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
11	2, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}
12		fdm 5367	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
13	12	imaeq2d 4966	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14		imadmrn 4976	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
15	13, 14	eqtr3di 2225	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
16	11, 15	eqtr3id 2224	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
17	16	eqeq1d 2186	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
18	17	exbidv 1825	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
19	9, 18	bitrid 192	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
20	8, 19	mpbid 147	. . . . . 6 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦ran 𝑓 = {𝑦})
21		vex 2740	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	snid 3622	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑦}
23		eleq2 2241	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
24	22, 23	mpbiri 168	. . . . . . . . 9 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25		frn 5370	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3154	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓 → 𝑦 ∈ 𝐴))
27	24, 26	syl5 32	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦 ∈ 𝐴))
28		dffn4 5440	. . . . . . . . . . . 12 ⊢ (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
29	5, 28	sylib 122	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}–onto→ran 𝑓)
30		fof 5434	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}–onto→ran 𝑓 → 𝑓:{𝐵}⟶ran 𝑓)
31	29, 30	syl 14	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓:{𝐵}⟶ran 𝑓)
32		feq3 5346	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓 ↔ 𝑓:{𝐵}⟶{𝑦}))
33	31, 32	syl5ibcom 155	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
34	2, 21	fsn 5684	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
35	33, 34	syl6ib 161	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓 = {⟨𝐵, 𝑦⟩}))
36	27, 35	jcad 307	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
37	36	eximdv 1880	. . . . . 6 ⊢ (𝑓:{𝐵}⟶𝐴 → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩})))
38	20, 37	mpd 13	. . . . 5 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
39		df-rex 2461	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐵, 𝑦⟩}))
40	38, 39	sylibr 134	. . . 4 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
41	2, 21	f1osn 5497	. . . . . . . . 9 ⊢ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42		f1oeq1 5445	. . . . . . . . 9 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
43	41, 42	mpbiri 168	. . . . . . . 8 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}–1-1-onto→{𝑦})
44		f1of 5457	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
45	43, 44	syl 14	. . . . . . 7 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46		snssi 3735	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
47		fss 5373	. . . . . . 7 ⊢ ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
48	45, 46, 47	syl2an 289	. . . . . 6 ⊢ ((𝑓 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 ∈ 𝐴) → 𝑓:{𝐵}⟶𝐴)
49	48	expcom 116	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
50	49	rexlimiv 2588	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)
51	40, 50	impbii 126	. . 3 ⊢ (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
52	4, 51	bitri 184	. 2 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
53	52	abbi2i 2292	1 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492  ∃!weu 2026   ∈ wcel 2148  {cab 2163  ∃wrex 2456  Vcvv 2737   ⊆ wss 3129  {csn 3591  ⟨cop 3594   class class class wbr 4000  dom cdm 4623  ran crn 4624   “ cima 4626   Fn wfn 5207  ⟶wf 5208  –onto→wfo 5210  –1-1-onto→wf1o 5211  (class class class)co 5869   ↑_𝑚 cmap 6642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-map 6644

This theorem is referenced by:  mapsnen  6805