Theorem map1 6986

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 6823	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		1oex 6589	. . 3 ⊢ 1o ∈ V
3		elex 2814	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		fnovex 6050	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 1o ∈ V ∧ 𝐴 ∈ V) → (1o ↑𝑚 𝐴) ∈ V)
5	1, 2, 3, 4	mp3an12i 1377	. 2 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ∈ V)
6	2	a1i 9	. 2 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V)
7		0ex 4216	. . 3 ⊢ ∅ ∈ V
8	7	2a1i 27	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1o ↑𝑚 𝐴) → ∅ ∈ V))
9		p0ex 4278	. . . 4 ⊢ {∅} ∈ V
10		xpexg 4840	. . . 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 6604	. . . . 5 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
14	13	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1o ↔ 𝑦 = ∅))
15		df1o2 6595	. . . . . . . 8 ⊢ 1o = {∅}
16	15	oveq1i 6027	. . . . . . 7 ⊢ (1o ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
17	16	eleq2i 2298	. . . . . 6 ⊢ (𝑥 ∈ (1o ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
18		elmapg 6829	. . . . . . 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 5870	. . . . 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 6940	1 ⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ∅c0 3494  {csn 3669   class class class wbr 4088   × cxp 4723   Fn wfn 5321  ⟶wf 5322  (class class class)co 6017  1_oc1o 6574   ↑_𝑚 cmap 6816   ≈ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-map 6818  df-en 6909

This theorem is referenced by: (None)