Theorem map2xp 9117

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 8445	. . . . 5 ⊢ 2o = {∅, 1o}
2		df-pr 4595	. . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o})
3	1, 2	eqtri 2753	. . . 4 ⊢ 2o = ({∅} ∪ {1o})
4	3	oveq2i 7401	. . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o}))
5		snex 5394	. . . . 5 ⊢ {∅} ∈ V
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
7		snex 5394	. . . . 5 ⊢ {1o} ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V)
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
10		1n0 8455	. . . . . . . 8 ⊢ 1o ≠ ∅
11	10	neii 2928	. . . . . . 7 ⊢ ¬ 1o = ∅
12		elsni 4609	. . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅)
13	11, 12	mto 197	. . . . . 6 ⊢ ¬ 1o ∈ {∅}
14		disjsn 4678	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅})
15	13, 14	mpbir 231	. . . . 5 ⊢ ({∅} ∩ {1o}) = ∅
16	15	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅)
17		mapunen 9116	. . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
18	6, 8, 9, 16, 17	syl31anc 1375	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
19	4, 18	eqbrtrid 5145	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})))
20		0ex 5265	. . . . 5 ⊢ ∅ ∈ V
21	20	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
22	9, 21	mapsnend 9010	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴)
23		1oex 8447	. . . . 5 ⊢ 1o ∈ V
24	23	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V)
25	9, 24	mapsnend 9010	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴)
26		xpen 9110	. . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
27	22, 25, 26	syl2anc 584	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴))
28		entr 8980	. 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 584	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ∩ cin 3916  ∅c0 4299  {csn 4592  {cpr 4594   class class class wbr 5110   × cxp 5639  (class class class)co 7390  1_oc1o 8430  2_oc2o 8431   ↑_m cmap 8802   ≈ cen 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923

This theorem is referenced by:  pwxpndom2  10625