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Theorem map2xp 8075
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 7519 . . . . 5 2𝑜 = {∅, 1𝑜}
2 df-pr 4156 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
31, 2eqtri 2648 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
43oveq2i 6616 . . 3 (𝐴𝑚 2𝑜) = (𝐴𝑚 ({∅} ∪ {1𝑜}))
5 snex 4874 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 4874 . . . . 5 {1𝑜} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1𝑜} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 7521 . . . . . . . 8 1𝑜 ≠ ∅
1110neii 2798 . . . . . . 7 ¬ 1𝑜 = ∅
12 elsni 4170 . . . . . . 7 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
1311, 12mto 188 . . . . . 6 ¬ 1𝑜 ∈ {∅}
14 disjsn 4221 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
1513, 14mpbir 221 . . . . 5 ({∅} ∩ {1𝑜}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17 mapunen 8074 . . . 4 ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
186, 8, 9, 16, 17syl31anc 1326 . . 3 (𝐴𝑉 → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
194, 18syl5eqbr 4653 . 2 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
20 oveq1 6612 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {∅}) = (𝐴𝑚 {∅}))
21 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
2220, 21breq12d 4631 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {∅}) ≈ 𝑥 ↔ (𝐴𝑚 {∅}) ≈ 𝐴))
23 vex 3194 . . . . 5 𝑥 ∈ V
24 0ex 4755 . . . . 5 ∅ ∈ V
2523, 24mapsnen 7980 . . . 4 (𝑥𝑚 {∅}) ≈ 𝑥
2622, 25vtoclg 3257 . . 3 (𝐴𝑉 → (𝐴𝑚 {∅}) ≈ 𝐴)
27 oveq1 6612 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {1𝑜}) = (𝐴𝑚 {1𝑜}))
2827, 21breq12d 4631 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴𝑚 {1𝑜}) ≈ 𝐴))
29 df1o2 7518 . . . . . 6 1𝑜 = {∅}
3029, 5eqeltri 2700 . . . . 5 1𝑜 ∈ V
3123, 30mapsnen 7980 . . . 4 (𝑥𝑚 {1𝑜}) ≈ 𝑥
3228, 31vtoclg 3257 . . 3 (𝐴𝑉 → (𝐴𝑚 {1𝑜}) ≈ 𝐴)
33 xpen 8068 . . 3 (((𝐴𝑚 {∅}) ≈ 𝐴 ∧ (𝐴𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
3426, 32, 33syl2anc 692 . 2 (𝐴𝑉 → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35 entr 7953 . 2 (((𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ∧ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴)) → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
3619, 34, 35syl2anc 692 1 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  cin 3559  c0 3896  {csn 4153  {cpr 4155   class class class wbr 4618   × cxp 5077  (class class class)co 6605  1𝑜c1o 7499  2𝑜c2o 7500  𝑚 cmap 7803  cen 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-1o 7506  df-2o 7507  df-er 7688  df-map 7805  df-en 7901  df-dom 7902
This theorem is referenced by:  pwxpndom2  9432
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