Theorem mapsn 6790

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 4237	. . . 4 ⊢ {𝐵} ∈ V
4	1, 3	elmap 6777	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5		ffn 5435	. . . . . . . 8 ⊢ (𝑓:{𝐵}⟶𝐴 → 𝑓 Fn {𝐵})
6	2	snid 3669	. . . . . . . 8 ⊢ 𝐵 ∈ {𝐵}
7		fneu 5389	. . . . . . . 8 ⊢ ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
8	5, 6, 7	sylancl 413	. . . . . . 7 ⊢ (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9		euabsn 3708	. . . . . . . 8 ⊢ (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦 ∣ 𝐵𝑓𝑦} = {𝑦})
10		imasng 5056	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦})
11	2, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑓 “ {𝐵}) = {𝑦 ∣ 𝐵𝑓𝑦}
12		fdm 5441	. . . . . . . . . . . . 13 ⊢ (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
13	12	imaeq2d 5031	. . . . . . . . . . . 12 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14		imadmrn 5041	. . . . . . . . . . . 12 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
15	13, 14	eqtr3di 2254	. . . . . . . . . . 11 ⊢ (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
16	11, 15	eqtr3id 2253	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → {𝑦 ∣ 𝐵𝑓𝑦} = ran 𝑓)
17	16	eqeq1d 2215	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶𝐴 → ({𝑦 ∣ 𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
18	17	exbidv 1849	. . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	snid 3669	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑦}
23		eleq2 2270	. . . . . . . . . 10 ⊢ (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
24	22, 23	mpbiri 168	. . . . . . . . 9 ⊢ (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25		frn 5444	. . . . . . . . . 10 ⊢ (𝑓:{𝐵}⟶𝐴 → ran 𝑓 ⊆ 𝐴)
26	25	sseld 3196	. . . . . . . . 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 5516	. . . . . . . . . . . 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 5510	. . . . . . . . . . 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 5420	. . . . . . . . . 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 5765	. . . . . . . . 9 ⊢ (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
35	33, 34	imbitrdi 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 1904	. . . . . 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 2491	. . . . 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 5575	. . . . . . . . 9 ⊢ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42		f1oeq1 5522	. . . . . . . . 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 5534	. . . . . . . 8 ⊢ (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
45	43, 44	syl 14	. . . . . . 7 ⊢ (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46		snssi 3783	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
47		fss 5447	. . . . . . 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 2618	. . . 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 2321	1 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  {cab 2192  ∃wrex 2486  Vcvv 2773   ⊆ wss 3170  {csn 3638  ⟨cop 3641   class class class wbr 4051  dom cdm 4683  ran crn 4684   “ cima 4686   Fn wfn 5275  ⟶wf 5276  –onto→wfo 5278  –1-1-onto→wf1o 5279  (class class class)co 5957   ↑_𝑚 cmap 6748

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-map 6750

This theorem is referenced by:  mapsnen  6917