MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  map2xp Structured version   Visualization version   GIF version

Theorem map2xp 8214
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 7661 . . . . 5 2𝑜 = {∅, 1𝑜}
2 df-pr 4256 . . . . 5 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
31, 2eqtri 2714 . . . 4 2𝑜 = ({∅} ∪ {1𝑜})
43oveq2i 6744 . . 3 (𝐴𝑚 2𝑜) = (𝐴𝑚 ({∅} ∪ {1𝑜}))
5 snex 4981 . . . . 5 {∅} ∈ V
65a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
7 snex 4981 . . . . 5 {1𝑜} ∈ V
87a1i 11 . . . 4 (𝐴𝑉 → {1𝑜} ∈ V)
9 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
10 1n0 7663 . . . . . . . 8 1𝑜 ≠ ∅
1110neii 2866 . . . . . . 7 ¬ 1𝑜 = ∅
12 elsni 4270 . . . . . . 7 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
1311, 12mto 188 . . . . . 6 ¬ 1𝑜 ∈ {∅}
14 disjsn 4321 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ ↔ ¬ 1𝑜 ∈ {∅})
1513, 14mpbir 221 . . . . 5 ({∅} ∩ {1𝑜}) = ∅
1615a1i 11 . . . 4 (𝐴𝑉 → ({∅} ∩ {1𝑜}) = ∅)
17 mapunen 8213 . . . 4 ((({∅} ∈ V ∧ {1𝑜} ∈ V ∧ 𝐴𝑉) ∧ ({∅} ∩ {1𝑜}) = ∅) → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
186, 8, 9, 16, 17syl31anc 1410 . . 3 (𝐴𝑉 → (𝐴𝑚 ({∅} ∪ {1𝑜})) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
194, 18syl5eqbr 4763 . 2 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})))
20 oveq1 6740 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {∅}) = (𝐴𝑚 {∅}))
21 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
2220, 21breq12d 4741 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {∅}) ≈ 𝑥 ↔ (𝐴𝑚 {∅}) ≈ 𝐴))
23 vex 3275 . . . . 5 𝑥 ∈ V
24 0ex 4866 . . . . 5 ∅ ∈ V
2523, 24mapsnen 8119 . . . 4 (𝑥𝑚 {∅}) ≈ 𝑥
2622, 25vtoclg 3338 . . 3 (𝐴𝑉 → (𝐴𝑚 {∅}) ≈ 𝐴)
27 oveq1 6740 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑚 {1𝑜}) = (𝐴𝑚 {1𝑜}))
2827, 21breq12d 4741 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑚 {1𝑜}) ≈ 𝑥 ↔ (𝐴𝑚 {1𝑜}) ≈ 𝐴))
29 df1o2 7660 . . . . . 6 1𝑜 = {∅}
3029, 5eqeltri 2767 . . . . 5 1𝑜 ∈ V
3123, 30mapsnen 8119 . . . 4 (𝑥𝑚 {1𝑜}) ≈ 𝑥
3228, 31vtoclg 3338 . . 3 (𝐴𝑉 → (𝐴𝑚 {1𝑜}) ≈ 𝐴)
33 xpen 8207 . . 3 (((𝐴𝑚 {∅}) ≈ 𝐴 ∧ (𝐴𝑚 {1𝑜}) ≈ 𝐴) → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
3426, 32, 33syl2anc 696 . 2 (𝐴𝑉 → ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴))
35 entr 8092 . 2 (((𝐴𝑚 2𝑜) ≈ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ∧ ((𝐴𝑚 {∅}) × (𝐴𝑚 {1𝑜})) ≈ (𝐴 × 𝐴)) → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
3619, 34, 35syl2anc 696 1 (𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1564  wcel 2071  Vcvv 3272  cun 3646  cin 3647  c0 3991  {csn 4253  {cpr 4255   class class class wbr 4728   × cxp 5184  (class class class)co 6733  1𝑜c1o 7641  2𝑜c2o 7642  𝑚 cmap 7942  cen 8037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-1st 7253  df-2nd 7254  df-1o 7648  df-2o 7649  df-er 7830  df-map 7944  df-en 8041  df-dom 8042
This theorem is referenced by:  pwxpndom2  9568
  Copyright terms: Public domain W3C validator