Theorem mapsnen 6805

Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
mapsnen	⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴

Proof of Theorem mapsnen

Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6649	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		mapsnen.1	. . 3 ⊢ 𝐴 ∈ V
3		mapsnen.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	snex 4182	. . 3 ⊢ {𝐵} ∈ V
5		fnovex 5902	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴 ↑𝑚 {𝐵}) ∈ V)
6	1, 2, 4, 5	mp3an 1337	. 2 ⊢ (𝐴 ↑𝑚 {𝐵}) ∈ V
7		vex 2740	. . . 4 ⊢ 𝑧 ∈ V
8	7, 3	fvex 5531	. . 3 ⊢ (𝑧‘𝐵) ∈ V
9	8	a1i 9	. 2 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)
10		vex 2740	. . . . 5 ⊢ 𝑤 ∈ V
11	3, 10	opex 4226	. . . 4 ⊢ ⟨𝐵, 𝑤⟩ ∈ V
12	11	snex 4182	. . 3 ⊢ {⟨𝐵, 𝑤⟩} ∈ V
13	12	a1i 9	. 2 ⊢ (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
14	2, 3	mapsn 6684	. . . . . 6 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
15	14	abeq2i 2288	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
16	15	anbi1i 458	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
17		r19.41v 2633	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
18		df-rex 2461	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
19	16, 17, 18	3bitr2i 208	. . 3 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
20		fveq1 5510	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21		vex 2740	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	3, 21	fvsn 5707	. . . . . . . . . 10 ⊢ ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
23	20, 22	eqtrdi 2226	. . . . . . . . 9 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = 𝑦)
24	23	eqeq2d 2189	. . . . . . . 8 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
25		equcom 1706	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
26	24, 25	bitrdi 196	. . . . . . 7 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
27	26	pm5.32i 454	. . . . . 6 ⊢ ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
28	27	anbi2i 457	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
29		anass 401	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
30		ancom 266	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
31	28, 29, 30	3bitr2i 208	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
32	31	exbii 1605	. . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
33		eleq1w 2238	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
34		opeq2 3777	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
35	34	sneqd 3604	. . . . . 6 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
36	35	eqeq2d 2189	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
37	33, 36	anbi12d 473	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
38	10, 37	ceqsexv 2776	. . 3 ⊢ (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
39	19, 32, 38	3bitri 206	. 2 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
40	6, 2, 9, 13, 39	en2i 6764	1 ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  Vcvv 2737  {csn 3591  ⟨cop 3594   class class class wbr 4000   × cxp 4621   Fn wfn 5207  ‘cfv 5212  (class class class)co 5869   ↑_𝑚 cmap 6642   ≈ cen 6732

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-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  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-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  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-1st 6135  df-2nd 6136  df-map 6644  df-en 6735

This theorem is referenced by: (None)