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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 | opelxpi 5072 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) | |
2 | 1 | ancoms 468 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
3 | 2 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
4 | eqidd 2611 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) | |
5 | sneq 4135 | . . . . . . . . . 10 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → {𝑧} = {〈𝑦, 𝑥〉}) | |
6 | 5 | cnveqd 5220 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ◡{𝑧} = ◡{〈𝑦, 𝑥〉}) |
7 | 6 | unieqd 4382 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = ∪ ◡{〈𝑦, 𝑥〉}) |
8 | opswap 5540 | . . . . . . . 8 ⊢ ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉 | |
9 | 7, 8 | syl6eq 2660 | . . . . . . 7 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = 〈𝑥, 𝑦〉) |
10 | 9 | mpt2mpt 6650 | . . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) |
11 | 10 | eqcomi 2619 | . . . . 5 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) |
12 | 11 | a1i 11 | . . . 4 ⊢ (⊤ → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧})) |
13 | 3, 4, 12, 9 | fmpt2co 7147 | . . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉)) |
14 | 13 | trud 1484 | . 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
15 | id 22 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 = 〈𝑥, 𝑦〉) | |
16 | 15 | mpt2mpt 6650 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
17 | mptresid 5375 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌)) | |
18 | 14, 16, 17 | 3eqtr2i 2638 | 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 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {csn 4125 〈cop 4131 ∪ cuni 4372 ↦ cmpt 4643 I cid 4948 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: txswaphmeo 21418 |
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