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Theorem map2xp 9094
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 8421 . . . . 5 2o = {∅, 1o}
2 df-pr 4590 . . . . 5 {∅, 1o} = ({∅} ∪ {1o})
31, 2eqtri 2761 . . . 4 2o = ({∅} ∪ {1o})
43oveq2i 7369 . . 3 (𝐴m 2o) = (𝐴m ({∅} ∪ {1o}))
5 snex 5389 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 5389 . . . . 5 {1o} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1o} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 8435 . . . . . . . 8 1o ≠ ∅
1110neii 2942 . . . . . . 7 ¬ 1o = ∅
12 elsni 4604 . . . . . . 7 (1o ∈ {∅} → 1o = ∅)
1311, 12mto 196 . . . . . 6 ¬ 1o ∈ {∅}
14 disjsn 4673 . . . . . 6 (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅})
1513, 14mpbir 230 . . . . 5 ({∅} ∩ {1o}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1o}) = ∅)
17 mapunen 9093 . . . 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 5141 . 2 (𝐴𝑉 → (𝐴m 2o) ≈ ((𝐴m {∅}) × (𝐴m {1o})))
20 0ex 5265 . . . . 5 ∅ ∈ V
2120a1i 11 . . . 4 (𝐴𝑉 → ∅ ∈ V)
229, 21mapsnend 8983 . . 3 (𝐴𝑉 → (𝐴m {∅}) ≈ 𝐴)
23 1oex 8423 . . . . 5 1o ∈ V
2423a1i 11 . . . 4 (𝐴𝑉 → 1o ∈ V)
259, 24mapsnend 8983 . . 3 (𝐴𝑉 → (𝐴m {1o}) ≈ 𝐴)
26 xpen 9087 . . 3 (((𝐴m {∅}) ≈ 𝐴 ∧ (𝐴m {1o}) ≈ 𝐴) → ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴))
2722, 25, 26syl2anc 585 . 2 (𝐴𝑉 → ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴))
28 entr 8949 . 2 (((𝐴m 2o) ≈ ((𝐴m {∅}) × (𝐴m {1o})) ∧ ((𝐴m {∅}) × (𝐴m {1o})) ≈ (𝐴 × 𝐴)) → (𝐴m 2o) ≈ (𝐴 × 𝐴))
2919, 27, 28syl2anc 585 1 (𝐴𝑉 → (𝐴m 2o) ≈ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  cun 3909  cin 3910  c0 4283  {csn 4587  {cpr 4589   class class class wbr 5106   × cxp 5632  (class class class)co 7358  1oc1o 8406  2oc2o 8407  m cmap 8768  cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888
This theorem is referenced by:  pwxpndom2  10606
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