Theorem map2xp 8015

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 7460	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
2		df-pr 4128	. . . . 5 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
3	1, 2	eqtri 2632	. . . 4 ⊢ 2𝑜 = ({∅} ∪ {1𝑜})
4	3	oveq2i 6560	. . 3 ⊢ (𝐴 ↑𝑚 2𝑜) = (𝐴 ↑𝑚 ({∅} ∪ {1𝑜}))
5		snex 4835	. . . . 5 ⊢ {∅} ∈ V
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
7		snex 4835	. . . . 5 ⊢ {1𝑜} ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {1𝑜} ∈ V)
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
10		1n0 7462	. . . . . . . 8 ⊢ 1𝑜 ≠ ∅
11	10	neii 2784	. . . . . . 7 ⊢ ¬ 1𝑜 = ∅
12		elsni 4142	. . . . . . 7 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
13	11, 12	mto 187	. . . . . 6 ⊢ ¬ 1𝑜 ∈ {∅}
14		disjsn 4192	. . . . . 6 ⊢ (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
15	13, 14	mpbir 220	. . . . 5 ⊢ ({∅} ∩ {1𝑜}) = ∅
16	15	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17		mapunen 8014	. . . 4 ⊢ ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴 ↑𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
18	6, 8, 9, 16, 17	syl31anc 1321	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
19	4, 18	syl5eqbr 4618	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 2𝑜) ≈ ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})))
20		oveq1 6556	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 {∅}) = (𝐴 ↑𝑚 {∅}))
21		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
22	20, 21	breq12d 4596	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 {∅}) ≈ 𝑥 ↔ (𝐴 ↑𝑚 {∅}) ≈ 𝐴))
23		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
24		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
25	23, 24	mapsnen 7920	. . . 4 ⊢ (𝑥 ↑𝑚 {∅}) ≈ 𝑥
26	22, 25	vtoclg 3239	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 {∅}) ≈ 𝐴)
27		oveq1 6556	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ↑𝑚 {1𝑜}) = (𝐴 ↑𝑚 {1𝑜}))
28	27, 21	breq12d 4596	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ↑𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴))
29		df1o2 7459	. . . . . 6 ⊢ 1𝑜 = {∅}
30	29, 5	eqeltri 2684	. . . . 5 ⊢ 1𝑜 ∈ V
31	23, 30	mapsnen 7920	. . . 4 ⊢ (𝑥 ↑𝑚 {1𝑜}) ≈ 𝑥
32	28, 31	vtoclg 3239	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴)
33		xpen 8008	. . 3 ⊢ (((𝐴 ↑𝑚 {∅}) ≈ 𝐴 ∧ (𝐴 ↑𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
34	26, 32, 33	syl2anc 691	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑𝑚 {∅}) × (𝐴 ↑𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35		entr 7894	. 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 691	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 2𝑜) ≈ (𝐴 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583   × cxp 5036  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_𝑚 cmap 7744   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842  df-dom 7843

This theorem is referenced by:  pwxpndom2  9366