Theorem map1 7067

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)

Assertion

Ref	Expression
map1	⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o)

Proof of Theorem map1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6902	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		1oex 6668	. . 3 ⊢ 1o ∈ V
3		elex 2827	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		fnovex 6091	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o ↑𝑚 𝐴) ∈ V)
5	1, 2, 3, 4	mp3an12i 1378	. 2 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ∈ V)
6	2	a1i 9	. 2 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V)
7		0ex 4242	. . 3 ⊢ ∅ ∈ V
8	7	2a1i 27	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) → ∅ ∈ V))
9		p0ex 4306	. . . 4 ⊢ {∅} ∈ V
10		xpexg 4869	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
11	9, 10	mpan2 425	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V)
12	11	a1d 22	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o → (𝐴 × {∅}) ∈ V))
13		el1o 6683	. . . . 5 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	13	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o ↔ 𝑦 = ∅))
15		df1o2 6674	. . . . . . . 8 ⊢ 1o = {∅}
16	15	oveq1i 6068	. . . . . . 7 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
17	16	eleq2i 2301	. . . . . 6 ⊢ (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
18		elmapg 6908	. . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
19	9, 18	mpan 424	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
20	17, 19	bitrid 192	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
21	7	fconst2 5906	. . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
22	20, 21	bitr2di 197	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1o ↑𝑚 𝐴)))
23	14, 22	anbi12d 473	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴))))
24		ancom 266	. . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1o ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅))
25	23, 24	bitr2di 197	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1o ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1o ∧ 𝑥 = (𝐴 × {∅}))))
26	5, 6, 8, 12, 25	en2d 7020	1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ∅c0 3512  {csn 3694   class class class wbr 4114   × cxp 4752   Fn wfn 5352  ⟶wf 5353  (class class class)co 6058  1_oc1o 6653   ↑_𝑚 cmap 6895   ≈ cen 6986

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-map 6897  df-en 6989

This theorem is referenced by: (None)