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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 8106 | . . . . 5 ⊢ 2o = {∅, 1o} | |
2 | df-pr 4560 | . . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
3 | 1, 2 | eqtri 2841 | . . . 4 ⊢ 2o = ({∅} ∪ {1o}) |
4 | 3 | oveq2i 7156 | . . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o})) |
5 | snex 5322 | . . . . 5 ⊢ {∅} ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
7 | snex 5322 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
10 | 1n0 8108 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
11 | 10 | neii 3015 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
12 | elsni 4574 | . . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅) | |
13 | 11, 12 | mto 198 | . . . . . 6 ⊢ ¬ 1o ∈ {∅} |
14 | disjsn 4639 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅}) | |
15 | 13, 14 | mpbir 232 | . . . . 5 ⊢ ({∅} ∩ {1o}) = ∅ |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅) |
17 | mapunen 8674 | . . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) | |
18 | 6, 8, 9, 16, 17 | syl31anc 1365 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
19 | 4, 18 | eqbrtrid 5092 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
20 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
22 | 9, 21 | mapsnend 8576 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴) |
23 | 1oex 8099 | . . . . 5 ⊢ 1o ∈ V | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
25 | 9, 24 | mapsnend 8576 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴) |
26 | xpen 8668 | . . 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 8549 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4557 {cpr 4559 class class class wbr 5057 × cxp 5546 (class class class)co 7145 1oc1o 8084 2oc2o 8085 ↑m cmap 8395 ≈ cen 8494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-1o 8091 df-2o 8092 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 |
This theorem is referenced by: pwxpndom2 10075 |
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