Theorem map2xp 9085

Description: A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 17-Jul-2022.)

Assertion

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

Proof of Theorem map2xp

Step	Hyp	Ref	Expression
1		df2o3 8413	. . . . 5 ⊢ 2o = {∅, 1o}
2		df-pr 4570	. . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o})
3	1, 2	eqtri 2759	. . . 4 ⊢ 2o = ({∅} ∪ {1o})
4	3	oveq2i 7378	. . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o}))
5		snex 5381	. . . . 5 ⊢ {∅} ∈ V
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
7		snex 5381	. . . . 5 ⊢ {1o} ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V)
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
10		1n0 8423	. . . . . . . 8 ⊢ 1o ≠ ∅
11	10	neii 2934	. . . . . . 7 ⊢ ¬ 1o = ∅
12		elsni 4584	. . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅)
13	11, 12	mto 197	. . . . . 6 ⊢ ¬ 1o ∈ {∅}
14		disjsn 4655	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅})
15	13, 14	mpbir 231	. . . . 5 ⊢ ({∅} ∩ {1o}) = ∅
16	15	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅)
17		mapunen 9084	. . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
18	6, 8, 9, 16, 17	syl31anc 1376	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
19	4, 18	eqbrtrid 5120	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
20		0ex 5242	. . . . 5 ⊢ ∅ ∈ V
21	20	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
22	9, 21	mapsnend 8983	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴)
23		1oex 8415	. . . . 5 ⊢ 1o ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V)
25	9, 24	mapsnend 8983	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴)
26		xpen 9078	. . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
27	22, 25, 26	syl2anc 585	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
28		entr 8953	. 2 ⊢ (((𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ∧ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
29	19, 27, 28	syl2anc 585	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ∩ cin 3888  ∅c0 4273  {csn 4567  {cpr 4569   class class class wbr 5085   × cxp 5629  (class class class)co 7367  1_oc1o 8398  2_oc2o 8399   ↑_m cmap 8773   ≈ cen 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895

This theorem is referenced by:  pwxpndom2  10588