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Theorem map2xp 8171
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 7618 . . . . 5 2𝑜 = {∅, 1𝑜}
2 df-pr 4213 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
31, 2eqtri 2673 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
43oveq2i 6701 . . 3 (𝐴𝑚 2𝑜) = (𝐴𝑚 ({∅} ∪ {1𝑜}))
5 snex 4938 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 4938 . . . . 5 {1𝑜} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1𝑜} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 7620 . . . . . . . 8 1𝑜 ≠ ∅
1110neii 2825 . . . . . . 7 ¬ 1𝑜 = ∅
12 elsni 4227 . . . . . . 7 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
1311, 12mto 188 . . . . . 6 ¬ 1𝑜 ∈ {∅}
14 disjsn 4278 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
1513, 14mpbir 221 . . . . 5 ({∅} ∩ {1𝑜}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17 mapunen 8170 . . . 4 ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
186, 8, 9, 16, 17syl31anc 1369 . . 3 (𝐴𝑉 → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
194, 18syl5eqbr 4720 . 2 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
20 oveq1 6697 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {∅}) = (𝐴𝑚 {∅}))
21 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
2220, 21breq12d 4698 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {∅}) ≈ 𝑥 ↔ (𝐴𝑚 {∅}) ≈ 𝐴))
23 vex 3234 . . . . 5 𝑥 ∈ V
24 0ex 4823 . . . . 5 ∅ ∈ V
2523, 24mapsnen 8076 . . . 4 (𝑥𝑚 {∅}) ≈ 𝑥
2622, 25vtoclg 3297 . . 3 (𝐴𝑉 → (𝐴𝑚 {∅}) ≈ 𝐴)
27 oveq1 6697 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {1𝑜}) = (𝐴𝑚 {1𝑜}))
2827, 21breq12d 4698 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴𝑚 {1𝑜}) ≈ 𝐴))
29 df1o2 7617 . . . . . 6 1𝑜 = {∅}
3029, 5eqeltri 2726 . . . . 5 1𝑜 ∈ V
3123, 30mapsnen 8076 . . . 4 (𝑥𝑚 {1𝑜}) ≈ 𝑥
3228, 31vtoclg 3297 . . 3 (𝐴𝑉 → (𝐴𝑚 {1𝑜}) ≈ 𝐴)
33 xpen 8164 . . 3 (((𝐴𝑚 {∅}) ≈ 𝐴 ∧ (𝐴𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
3426, 32, 33syl2anc 694 . 2 (𝐴𝑉 → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35 entr 8049 . 2 (((𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ∧ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴)) → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
3619, 34, 35syl2anc 694 1 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212   class class class wbr 4685   × cxp 5141  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599  𝑚 cmap 7899  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-2o 7606  df-er 7787  df-map 7901  df-en 7998  df-dom 7999
This theorem is referenced by:  pwxpndom2  9525
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