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Mirrors > Home > MPE Home > Th. List > map2xp | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
map2xp | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2o3 8119 | . . . . 5 ⊢ 2o = {∅, 1o} | |
2 | df-pr 4572 | . . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
3 | 1, 2 | eqtri 2846 | . . . 4 ⊢ 2o = ({∅} ∪ {1o}) |
4 | 3 | oveq2i 7169 | . . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o})) |
5 | snex 5334 | . . . . 5 ⊢ {∅} ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
7 | snex 5334 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
10 | 1n0 8121 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
11 | 10 | neii 3020 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
12 | elsni 4586 | . . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅) | |
13 | 11, 12 | mto 199 | . . . . . 6 ⊢ ¬ 1o ∈ {∅} |
14 | disjsn 4649 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅}) | |
15 | 13, 14 | mpbir 233 | . . . . 5 ⊢ ({∅} ∩ {1o}) = ∅ |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅) |
17 | mapunen 8688 | . . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) | |
18 | 6, 8, 9, 16, 17 | syl31anc 1369 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
19 | 4, 18 | eqbrtrid 5103 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
20 | 0ex 5213 | . . . . 5 ⊢ ∅ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
22 | 9, 21 | mapsnend 8590 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴) |
23 | 1oex 8112 | . . . . 5 ⊢ 1o ∈ V | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
25 | 9, 24 | mapsnend 8590 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴) |
26 | xpen 8682 | . . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) | |
27 | 22, 25, 26 | syl2anc 586 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) |
28 | entr 8563 | . 2 ⊢ (((𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ∧ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) | |
29 | 19, 27, 28 | syl2anc 586 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 class class class wbr 5068 × cxp 5555 (class class class)co 7158 1oc1o 8097 2oc2o 8098 ↑m cmap 8408 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-2o 8105 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 |
This theorem is referenced by: pwxpndom2 10089 |
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