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Mirrors > Home > ILE Home > Th. List > txswaphmeolem | GIF version |
Description: Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
txswaphmeolem | ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = ( I ↾ (𝑋 × 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4643 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) | |
2 | 1 | ancoms 266 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
3 | 2 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
4 | eqidd 2171 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) | |
5 | sneq 3594 | . . . . . . . . . 10 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → {𝑧} = {〈𝑦, 𝑥〉}) | |
6 | 5 | cnveqd 4787 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ◡{𝑧} = ◡{〈𝑦, 𝑥〉}) |
7 | 6 | unieqd 3807 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = ∪ ◡{〈𝑦, 𝑥〉}) |
8 | vex 2733 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
9 | vex 2733 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
10 | opswapg 5097 | . . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉) | |
11 | 8, 9, 10 | mp2an 424 | . . . . . . . 8 ⊢ ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉 |
12 | 7, 11 | eqtrdi 2219 | . . . . . . 7 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = 〈𝑥, 𝑦〉) |
13 | 12 | mpompt 5945 | . . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) |
14 | 13 | eqcomi 2174 | . . . . 5 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) |
15 | 14 | a1i 9 | . . . 4 ⊢ (⊤ → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧})) |
16 | 3, 4, 15, 12 | fmpoco 6195 | . . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉)) |
17 | 16 | mptru 1357 | . 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
18 | id 19 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 = 〈𝑥, 𝑦〉) | |
19 | 18 | mpompt 5945 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
20 | mptresid 4945 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌)) | |
21 | 17, 19, 20 | 3eqtr2i 2197 | 1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = ( I ↾ (𝑋 × 𝑌)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ⊤wtru 1349 ∈ wcel 2141 Vcvv 2730 {csn 3583 〈cop 3586 ∪ cuni 3796 ↦ cmpt 4050 I cid 4273 × cxp 4609 ◡ccnv 4610 ↾ cres 4613 ∘ ccom 4615 ∈ cmpo 5855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 |
This theorem is referenced by: txswaphmeo 13115 |
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