Theorem mapsnend 7043

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.) (Revised by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
mapsnend.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
mapsnend.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
mapsnend	⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴)

Proof of Theorem mapsnend

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

Step	Hyp	Ref	Expression
1		fnmap 6880	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		mapsnend.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	elexd 2826	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
4		mapsnend.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		snexg 4289	. . . 4 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
6	4, 5	syl 14	. . 3 ⊢ (𝜑 → {𝐵} ∈ V)
7		fnovex 6074	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴 ↑𝑚 {𝐵}) ∈ V)
8	1, 3, 6, 7	mp3an2i 1379	. 2 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ∈ V)
9		vex 2815	. . . 4 ⊢ 𝑧 ∈ V
10		fvexg 5680	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑧‘𝐵) ∈ V)
11	9, 4, 10	sylancr 414	. . 3 ⊢ (𝜑 → (𝑧‘𝐵) ∈ V)
12	11	a1d 22	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V))
13		vex 2815	. . . . 5 ⊢ 𝑤 ∈ V
14		opexg 4335	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ V) → ⟨𝐵, 𝑤⟩ ∈ V)
15	4, 13, 14	sylancl 413	. . . 4 ⊢ (𝜑 → ⟨𝐵, 𝑤⟩ ∈ V)
16		snexg 4289	. . . 4 ⊢ (⟨𝐵, 𝑤⟩ ∈ V → {⟨𝐵, 𝑤⟩} ∈ V)
17	15, 16	syl 14	. . 3 ⊢ (𝜑 → {⟨𝐵, 𝑤⟩} ∈ V)
18	17	a1d 22	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
19	2, 4	mapsnd 6914	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
20	19	eqabrd 2370	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
21	20	anbi1d 465	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
22		r19.41v 2699	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
23	22	bicomi 132	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
24	23	a1i 9	. . . 4 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
25		df-rex 2526	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
26	25	a1i 9	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))))
27	21, 24, 26	3bitrd 214	. . 3 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))))
28		fveq1 5660	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
29		vex 2815	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
30		fvsng 5871	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
31	4, 29, 30	sylancl 413	. . . . . . . . . 10 ⊢ (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
32	28, 31	sylan9eqr 2287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧‘𝐵) = 𝑦)
33	32	eqeq2d 2244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
34		equcom 1754	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
35	33, 34	bitrdi 196	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
36	35	pm5.32da 452	. . . . . 6 ⊢ (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
37	36	anbi2d 464	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
38		anass 401	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
39	38	a1i 9	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
40		ancom 266	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
41	40	a1i 9	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
42	37, 39, 41	3bitr2d 216	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
43	42	exbidv 1874	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}))))
44		eleq1w 2293	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
45		opeq2 3877	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
46	45	sneqd 3695	. . . . . . 7 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
47	46	eqeq2d 2244	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
48	44, 47	anbi12d 473	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
49	48	equsexvw 1777	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
50	49	a1i 9	. . 3 ⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
51	27, 43, 50	3bitrd 214	. 2 ⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
52	8, 2, 12, 18, 51	en2d 6998	1 ⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  Vcvv 2812  {csn 3682  ⟨cop 3685   class class class wbr 4102   × cxp 4738   Fn wfn 5338  ‘cfv 5343  (class class class)co 6041   ↑_𝑚 cmap 6873   ≈ cen 6964

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-ov 6044  df-oprab 6045  df-mpo 6046  df-1st 6325  df-2nd 6326  df-map 6875  df-en 6967

This theorem is referenced by:  mapfi  7205