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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 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))

Proof of Theorem map2xp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df2o3 7460 . . . . 5 2𝑜 = {∅, 1𝑜}
2 df-pr 4128 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
31, 2eqtri 2632 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
43oveq2i 6560 . . 3 (𝐴𝑚 2𝑜) = (𝐴𝑚 ({∅} ∪ {1𝑜}))
5 snex 4835 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 4835 . . . . 5 {1𝑜} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1𝑜} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 7462 . . . . . . . 8 1𝑜 ≠ ∅
1110neii 2784 . . . . . . 7 ¬ 1𝑜 = ∅
12 elsni 4142 . . . . . . 7 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
1311, 12mto 187 . . . . . 6 ¬ 1𝑜 ∈ {∅}
14 disjsn 4192 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
1513, 14mpbir 220 . . . . 5 ({∅} ∩ {1𝑜}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17 mapunen 8014 . . . 4 ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
186, 8, 9, 16, 17syl31anc 1321 . . 3 (𝐴𝑉 → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
194, 18syl5eqbr 4618 . 2 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
20 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {∅}) = (𝐴𝑚 {∅}))
21 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
2220, 21breq12d 4596 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {∅}) ≈ 𝑥 ↔ (𝐴𝑚 {∅}) ≈ 𝐴))
23 vex 3176 . . . . 5 𝑥 ∈ V
24 0ex 4718 . . . . 5 ∅ ∈ V
2523, 24mapsnen 7920 . . . 4 (𝑥𝑚 {∅}) ≈ 𝑥
2622, 25vtoclg 3239 . . 3 (𝐴𝑉 → (𝐴𝑚 {∅}) ≈ 𝐴)
27 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {1𝑜}) = (𝐴𝑚 {1𝑜}))
2827, 21breq12d 4596 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴𝑚 {1𝑜}) ≈ 𝐴))
29 df1o2 7459 . . . . . 6 1𝑜 = {∅}
3029, 5eqeltri 2684 . . . . 5 1𝑜 ∈ V
3123, 30mapsnen 7920 . . . 4 (𝑥𝑚 {1𝑜}) ≈ 𝑥
3228, 31vtoclg 3239 . . 3 (𝐴𝑉 → (𝐴𝑚 {1𝑜}) ≈ 𝐴)
33 xpen 8008 . . 3 (((𝐴𝑚 {∅}) ≈ 𝐴 ∧ (𝐴𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
3426, 32, 33syl2anc 691 . 2 (𝐴𝑉 → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35 entr 7894 . 2 (((𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ∧ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴)) → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
3619, 34, 35syl2anc 691 1 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
 Colors of variables: wff setvar class 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  1𝑜c1o 7440  2𝑜c2o 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
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