Theorem map2xp 9213

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 8530	. . . . 5 ⊢ 2o = {∅, 1o}
2		df-pr 4651	. . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o})
3	1, 2	eqtri 2768	. . . 4 ⊢ 2o = ({∅} ∪ {1o})
4	3	oveq2i 7459	. . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o}))
5		snex 5451	. . . . 5 ⊢ {∅} ∈ V
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
7		snex 5451	. . . . 5 ⊢ {1o} ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V)
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
10		1n0 8544	. . . . . . . 8 ⊢ 1o ≠ ∅
11	10	neii 2948	. . . . . . 7 ⊢ ¬ 1o = ∅
12		elsni 4665	. . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅)
13	11, 12	mto 197	. . . . . 6 ⊢ ¬ 1o ∈ {∅}
14		disjsn 4736	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅})
15	13, 14	mpbir 231	. . . . 5 ⊢ ({∅} ∩ {1o}) = ∅
16	15	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅)
17		mapunen 9212	. . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
18	6, 8, 9, 16, 17	syl31anc 1373	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
19	4, 18	eqbrtrid 5201	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
20		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
21	20	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
22	9, 21	mapsnend 9101	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴)
23		1oex 8532	. . . . 5 ⊢ 1o ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V)
25	9, 24	mapsnend 9101	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴)
26		xpen 9206	. . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
27	22, 25, 26	syl2anc 583	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
28		entr 9066	. 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 583	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648  {cpr 4650   class class class wbr 5166   × cxp 5698  (class class class)co 7448  1_oc1o 8515  2_oc2o 8516   ↑_m cmap 8884   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005

This theorem is referenced by:  pwxpndom2  10734