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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 8421 | . . . . 5 ⊢ 2o = {∅, 1o} | |
2 | df-pr 4590 | . . . . 5 ⊢ {∅, 1o} = ({∅} ∪ {1o}) | |
3 | 1, 2 | eqtri 2761 | . . . 4 ⊢ 2o = ({∅} ∪ {1o}) |
4 | 3 | oveq2i 7369 | . . 3 ⊢ (𝐴 ↑m 2o) = (𝐴 ↑m ({∅} ∪ {1o})) |
5 | snex 5389 | . . . . 5 ⊢ {∅} ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
7 | snex 5389 | . . . . 5 ⊢ {1o} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {1o} ∈ V) |
9 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
10 | 1n0 8435 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
11 | 10 | neii 2942 | . . . . . . 7 ⊢ ¬ 1o = ∅ |
12 | elsni 4604 | . . . . . . 7 ⊢ (1o ∈ {∅} → 1o = ∅) | |
13 | 11, 12 | mto 196 | . . . . . 6 ⊢ ¬ 1o ∈ {∅} |
14 | disjsn 4673 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ ↔ ¬ 1o ∈ {∅}) | |
15 | 13, 14 | mpbir 230 | . . . . 5 ⊢ ({∅} ∩ {1o}) = ∅ |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({∅} ∩ {1o}) = ∅) |
17 | mapunen 9093 | . . . 4 ⊢ ((({∅} ∈ V ∧ {1o} ∈ V ∧ 𝐴 ∈ 𝑉) ∧ ({∅} ∩ {1o}) = ∅) → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) | |
18 | 6, 8, 9, 16, 17 | syl31anc 1374 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ({∅} ∪ {1o})) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
19 | 4, 18 | eqbrtrid 5141 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o}))) |
20 | 0ex 5265 | . . . . 5 ⊢ ∅ ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
22 | 9, 21 | mapsnend 8983 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {∅}) ≈ 𝐴) |
23 | 1oex 8423 | . . . . 5 ⊢ 1o ∈ V | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 1o ∈ V) |
25 | 9, 24 | mapsnend 8983 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {1o}) ≈ 𝐴) |
26 | xpen 9087 | . . 3 ⊢ (((𝐴 ↑m {∅}) ≈ 𝐴 ∧ (𝐴 ↑m {1o}) ≈ 𝐴) → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) | |
27 | 22, 25, 26 | syl2anc 585 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑m {∅}) × (𝐴 ↑m {1o})) ≈ (𝐴 × 𝐴)) |
28 | entr 8949 | . 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 585 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 {csn 4587 {cpr 4589 class class class wbr 5106 × cxp 5632 (class class class)co 7358 1oc1o 8406 2oc2o 8407 ↑m cmap 8768 ≈ cen 8883 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 |
This theorem is referenced by: pwxpndom2 10606 |
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