Theorem mapsnen 6903

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 6742	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		mapsnen.1	. . 3 ⊢ 𝐴 ∈ V
3		mapsnen.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	snex 4229	. . 3 ⊢ {𝐵} ∈ V
5		fnovex 5977	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴 ↑𝑚 {𝐵}) ∈ V)
6	1, 2, 4, 5	mp3an 1350	. 2 ⊢ (𝐴 ↑𝑚 {𝐵}) ∈ V
7		vex 2775	. . . 4 ⊢ 𝑧 ∈ V
8	7, 3	fvex 5596	. . 3 ⊢ (𝑧‘𝐵) ∈ V
9	8	a1i 9	. 2 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)
10		vex 2775	. . . . 5 ⊢ 𝑤 ∈ V
11	3, 10	opex 4273	. . . 4 ⊢ ⟨𝐵, 𝑤⟩ ∈ V
12	11	snex 4229	. . 3 ⊢ {⟨𝐵, 𝑤⟩} ∈ V
13	12	a1i 9	. 2 ⊢ (𝑤 ∈ 𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
14	2, 3	mapsn 6777	. . . . . 6 ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
15	14	abeq2i 2316	. . . . 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 2662	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵)))
18		df-rex 2490	. . . 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 5575	. . . . . . . . . 10 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21		vex 2775	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	3, 21	fvsn 5779	. . . . . . . . . 10 ⊢ ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
23	20, 22	eqtrdi 2254	. . . . . . . . 9 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧‘𝐵) = 𝑦)
24	23	eqeq2d 2217	. . . . . . . 8 ⊢ (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦))
25		equcom 1729	. . . . . . . 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 1628	. . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {⟨𝐵, 𝑦⟩})))
33		eleq1w 2266	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
34		opeq2 3820	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
35	34	sneqd 3646	. . . . . 6 ⊢ (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
36	35	eqeq2d 2217	. . . . 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 2811	. . 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 6861	1 ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∃wrex 2485  Vcvv 2772  {csn 3633  ⟨cop 3636   class class class wbr 4044   × cxp 4673   Fn wfn 5266  ‘cfv 5271  (class class class)co 5944   ↑_𝑚 cmap 6735   ≈ cen 6825

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-en 6828

This theorem is referenced by:  exmidpw2en  7009