Theorem map2xp 8130

Description: A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Assertion

Ref	Expression
map2xp	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))

Proof of Theorem map2xp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2o3 7573	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
2		df-pr 4180	. . . . 5 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
3	1, 2	eqtri 2644	. . . 4 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
4	3	oveq2i 6661	. . 3 ⊢ (𝐴 ↑𝑚 2𝑜) = (𝐴 ↑𝑚 ({∅} ∪ {1𝑜}))
5		snex 4908	. . . . 5 ⊢ {∅} ∈ V
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
7		snex 4908	. . . . 5 ⊢ {1𝑜} ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {1𝑜} ∈ V)
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
10		1n0 7575	. . . . . . . 8 ⊢ 1𝑜 ≠ ∅
11	10	neii 2796	. . . . . . 7 ⊢ ¬ 1𝑜 = ∅
12		elsni 4194	. . . . . . 7 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
13	11, 12	mto 188	. . . . . 6 ⊢ ¬ 1𝑜 ∈ {∅}
14		disjsn 4246	. . . . . 6 ⊢ (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
15	13, 14	mpbir 221	. . . . 5 ⊢ ({∅} ∩ {1𝑜}) = ∅
16	15	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17		mapunen 8129	. . . 4 ⊢ ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴 ↑𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
18	6, 8, 9, 16, 17	syl31anc 1329	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
19	4, 18	syl5eqbr 4688	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 2𝑜) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
20		oveq1 6657	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 {∅}) = (𝐴 ↑𝑚 {∅}))
21		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
22	20, 21	breq12d 4666	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 {∅}) ≈ 𝑥 ↔ (𝐴 ↑𝑚 {∅}) ≈ 𝐴))
23		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
24		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
25	23, 24	mapsnen 8035	. . . 4 ⊢ (𝑥 ↑𝑚 {∅}) ≈ 𝑥
26	22, 25	vtoclg 3266	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 {∅}) ≈ 𝐴)
27		oveq1 6657	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 {1𝑜}) = (𝐴 ↑𝑚 {1𝑜}))
28	27, 21	breq12d 4666	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴))
29		df1o2 7572	. . . . . 6 ⊢ 1𝑜 = {∅}
30	29, 5	eqeltri 2697	. . . . 5 ⊢ 1𝑜 ∈ V
31	23, 30	mapsnen 8035	. . . 4 ⊢ (𝑥 ↑𝑚 {1𝑜}) ≈ 𝑥
32	28, 31	vtoclg 3266	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴)
33		xpen 8123	. . 3 ⊢ (((𝐴 ↑𝑚 {∅}) ≈ 𝐴 ∧ (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
34	26, 32, 33	syl2anc 693	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35		entr 8008	. 2 ⊢ (((𝐴 ↑𝑚 2𝑜) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ∧ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ≈ (𝐴 × 𝐴)) → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))
36	19, 34, 35	syl2anc 693	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  {cpr 4179   class class class wbr 4653   × cxp 5112  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957

This theorem is referenced by:  pwxpndom2  9487