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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 8305 | . . . . 5 ⊢ 2o = {∅, 1o} | |
2 | df-pr 4564 | . . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
3 | 1, 2 | eqtri 2766 | . . . 4 ⊢ 2o = ({∅} ∪ {1o}) |
4 | 3 | oveq2i 7286 | . . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o})) |
5 | snex 5354 | . . . . 5 ⊢ {∅} ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
7 | snex 5354 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
10 | 1n0 8318 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
11 | 10 | neii 2945 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
12 | elsni 4578 | . . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅) | |
13 | 11, 12 | mto 196 | . . . . . 6 ⊢ ¬ 1o ∈ {∅} |
14 | disjsn 4647 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅}) | |
15 | 13, 14 | mpbir 230 | . . . . 5 ⊢ ({∅} ∩ {1o}) = ∅ |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅) |
17 | mapunen 8933 | . . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) | |
18 | 6, 8, 9, 16, 17 | syl31anc 1372 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
19 | 4, 18 | eqbrtrid 5109 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
20 | 0ex 5231 | . . . . 5 ⊢ ∅ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
22 | 9, 21 | mapsnend 8826 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴) |
23 | 1oex 8307 | . . . . 5 ⊢ 1o ∈ V | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
25 | 9, 24 | mapsnend 8826 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴) |
26 | xpen 8927 | . . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) | |
27 | 22, 25, 26 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) |
28 | entr 8792 | . 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 584 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 {csn 4561 {cpr 4563 class class class wbr 5074 × cxp 5587 (class class class)co 7275 1oc1o 8290 2oc2o 8291 ↑m cmap 8615 ≈ cen 8730 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 |
This theorem is referenced by: pwxpndom2 10421 |
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