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Mirrors > Home > MPE Home > Th. List > txswaphmeolem | Structured version Visualization version 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 | id 22 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 = 〈𝑥, 𝑦〉) | |
2 | 1 | mpompt 7547 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
3 | mptresid 6071 | . 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) | |
4 | opelxpi 5726 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) | |
5 | 4 | ancoms 458 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
7 | eqidd 2736 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) | |
8 | sneq 4641 | . . . . . . . . . 10 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → {𝑧} = {〈𝑦, 𝑥〉}) | |
9 | 8 | cnveqd 5889 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ◡{𝑧} = ◡{〈𝑦, 𝑥〉}) |
10 | 9 | unieqd 4925 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = ∪ ◡{〈𝑦, 𝑥〉}) |
11 | opswap 6251 | . . . . . . . 8 ⊢ ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉 | |
12 | 10, 11 | eqtrdi 2791 | . . . . . . 7 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = 〈𝑥, 𝑦〉) |
13 | 12 | mpompt 7547 | . . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) |
14 | 13 | eqcomi 2744 | . . . . 5 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) |
15 | 14 | a1i 11 | . . . 4 ⊢ (⊤ → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧})) |
16 | 6, 7, 15, 12 | fmpoco 8119 | . . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉)) |
17 | 16 | mptru 1544 | . 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
18 | 2, 3, 17 | 3eqtr4ri 2774 | 1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = ( I ↾ (𝑋 × 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 {csn 4631 〈cop 4637 ∪ cuni 4912 ↦ cmpt 5231 I cid 5582 × cxp 5687 ◡ccnv 5688 ↾ cres 5691 ∘ ccom 5693 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: txswaphmeo 23829 |
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