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Theorem map2xp 8749
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 (𝐴𝑉 → (𝐴m 2o) ≈ (𝐴 × 𝐴))

Proof of Theorem map2xp
StepHypRef Expression
1 df2o3 8158 . . . . 5 2o = {∅, 1o}
2 df-pr 4529 . . . . 5 {∅, 1o} = ({∅} ∪ {1o})
31, 2eqtri 2762 . . . 4 2o = ({∅} ∪ {1o})
43oveq2i 7193 . . 3 (𝐴m 2o) = (𝐴m ({∅} ∪ {1o}))
5 snex 5308 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 5308 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 8162 . . . . . . . 8 1o ≠ ∅
1110neii 2937 . . . . . . 7 ¬ 1o = ∅
12 elsni 4543 . . . . . . 7 (1o ∈ {∅} → 1o = ∅)
1311, 12mto 200 . . . . . 6 ¬ 1o ∈ {∅}
14 disjsn 4612 . . . . . 6 (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅})
1513, 14mpbir 234 . . . . 5 ({∅} ∩ {1o}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1o}) = ∅)
17 mapunen 8748 . . . 4 ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴m ({∅} ∪ {1o})) ≈ ((𝐴m {∅}) × (𝐴m {1o})))
186, 8, 9, 16, 17syl31anc 1374 . . 3 (𝐴𝑉 → (𝐴m ({∅} ∪ {1o})) ≈ ((𝐴m {∅}) × (𝐴m {1o})))
194, 18eqbrtrid 5075 . 2 (𝐴𝑉 → (𝐴m 2o) ≈ ((𝐴m {∅}) × (𝐴m {1o})))
20 0ex 5185 . . . . 5 ∅ ∈ V
2120a1i 11 . . . 4 (𝐴𝑉 → ∅ ∈ V)
229, 21mapsnend 8647 . . 3 (𝐴𝑉 → (𝐴m {∅}) ≈ 𝐴)
23 1oex 8156 . . . . 5 1o ∈ V
2423a1i 11 . . . 4 (𝐴𝑉 → 1o ∈ V)
259, 24mapsnend 8647 . . 3 (𝐴𝑉 → (𝐴m {1o}) ≈ 𝐴)
26 xpen 8742 . . 3 (((𝐴m {∅}) ≈ 𝐴 ∧ (𝐴m {1o}) ≈ 𝐴) → ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴))
2722, 25, 26syl2anc 587 . 2 (𝐴𝑉 → ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴))
28 entr 8619 . 2 (((𝐴m 2o) ≈ ((𝐴m {∅}) × (𝐴m {1o})) ∧ ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴)) → (𝐴m 2o) ≈ (𝐴 × 𝐴))
2919, 27, 28syl2anc 587 1 (𝐴𝑉 → (𝐴m 2o) ≈ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2114  Vcvv 3400  cun 3851  cin 3852  c0 4221  {csn 4526  {cpr 4528   class class class wbr 5040   × cxp 5533  (class class class)co 7182  1oc1o 8136  2oc2o 8137  m cmap 8449  cen 8564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-1o 8143  df-2o 8144  df-er 8332  df-map 8451  df-en 8568  df-dom 8569
This theorem is referenced by:  pwxpndom2  10177
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