Theorem mapsnen 6777

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 6621	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		mapsnen.1	. . 3 ⊢ 𝐴 ∈ V
3		mapsnen.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	snex 4164	. . 3 ⊢ {𝐵} ∈ V
5		fnovex 5875	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴 ↑𝑚 {𝐵}) ∈ V)
6	1, 2, 4, 5	mp3an 1327	. 2 ⊢ (𝐴 ↑𝑚 {𝐵}) ∈ V
7		vex 2729	. . . 4 ⊢ 𝑧 ∈ V
8	7, 3	fvex 5506	. . 3 ⊢ (𝑧‘𝐵) ∈ V
9	8	a1i 9	. 2 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)
10		vex 2729	. . . . 5 ⊢ 𝑤 ∈ V
11	3, 10	opex 4207	. . . 4 ⊢ ⟨𝐵, 𝑤⟩ ∈ V
12	11	snex 4164	. . 3 ⊢ {⟨𝐵, 𝑤⟩} ∈ V
13	12	a1i 9	. 2 ⊢ (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
14	2, 3	mapsn 6656	. . . . . 6 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
15	14	abeq2i 2277	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
16	15	anbi1i 454	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
17		r19.41v 2622	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
18		df-rex 2450	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
19	16, 17, 18	3bitr2i 207	. . 3 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
20		fveq1 5485	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21		vex 2729	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	3, 21	fvsn 5680	. . . . . . . . . 10 ⊢ ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
23	20, 22	eqtrdi 2215	. . . . . . . . 9 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = 𝑦)
24	23	eqeq2d 2177	. . . . . . . 8 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
25		equcom 1694	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
26	24, 25	bitrdi 195	. . . . . . 7 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
27	26	pm5.32i 450	. . . . . 6 ⊢ ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
28	27	anbi2i 453	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
29		anass 399	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
30		ancom 264	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
31	28, 29, 30	3bitr2i 207	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
32	31	exbii 1593	. . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
33		eleq1w 2227	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
34		opeq2 3759	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
35	34	sneqd 3589	. . . . . 6 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
36	35	eqeq2d 2177	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
37	33, 36	anbi12d 465	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
38	10, 37	ceqsexv 2765	. . 3 ⊢ (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
39	19, 32, 38	3bitri 205	. 2 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
40	6, 2, 9, 13, 39	en2i 6736	1 ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  Vcvv 2726  {csn 3576  ⟨cop 3579   class class class wbr 3982   × cxp 4602   Fn wfn 5183  ‘cfv 5188  (class class class)co 5842   ↑_𝑚 cmap 6614   ≈ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-en 6707

This theorem is referenced by: (None)