Theorem mapsnen 7920

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		ovex 6577	. 2 ⊢ (𝐴 ↑𝑚 {𝐵}) ∈ V
2		mapsnen.1	. 2 ⊢ 𝐴 ∈ V
3		fvex 6113	. . 3 ⊢ (𝑧‘𝐵) ∈ V
4	3	a1i 11	. 2 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)
5		snex 4835	. . 3 ⊢ {⟨𝐵, 𝑤⟩} ∈ V
6	5	a1i 11	. 2 ⊢ (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
7		mapsnen.2	. . . . . . 7 ⊢ 𝐵 ∈ V
8	2, 7	mapsn 7785	. . . . . 6 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
9	8	abeq2i 2722	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
10	9	anbi1i 727	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
11		r19.41v 3070	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
12		df-rex 2902	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
13	10, 11, 12	3bitr2i 287	. . 3 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))))
14		fveq1 6102	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
15		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
16	7, 15	fvsn 6351	. . . . . . . . . 10 ⊢ ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
17	14, 16	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = 𝑦)
18	17	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
19		equcom 1932	. . . . . . . 8 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
20	18, 19	syl6bb 275	. . . . . . 7 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))
21	20	pm5.32i 667	. . . . . 6 ⊢ ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
22	21	anbi2i 726	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
23		anass 679	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
24		ancom 465	. . . . 5 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
25	22, 23, 24	3bitr2i 287	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
26	25	exbii 1764	. . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
27		vex 3176	. . . 4 ⊢ 𝑤 ∈ V
28		eleq1 2676	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
29		opeq2 4341	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
30	29	sneqd 4137	. . . . . 6 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
31	30	eqeq2d 2620	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
32	28, 31	anbi12d 743	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩})))
33	27, 32	ceqsexv 3215	. . 3 ⊢ (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
34	13, 26, 33	3bitri 285	. 2 ⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑤⟩}))
35	1, 2, 4, 6, 34	en2i 7879	1 ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  {csn 4125  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-en 7842

This theorem is referenced by:  map2xp  8015  mapdom3  8017  ackbij1lem5  8929  pwxpndom2  9366  hashmap  13082