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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 6069 | . 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) | |
| 4 | opelxpi 5722 | . . . . . 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 2738 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) | |
| 8 | sneq 4636 | . . . . . . . . . 10 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → {𝑧} = {〈𝑦, 𝑥〉}) | |
| 9 | 8 | cnveqd 5886 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ◡{𝑧} = ◡{〈𝑦, 𝑥〉}) | 
| 10 | 9 | unieqd 4920 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = ∪ ◡{〈𝑦, 𝑥〉}) | 
| 11 | opswap 6249 | . . . . . . . 8 ⊢ ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉 | |
| 12 | 10, 11 | eqtrdi 2793 | . . . . . . 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 2746 | . . . . 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 8120 | . . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉)) | 
| 17 | 16 | mptru 1547 | . 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) | 
| 18 | 2, 3, 17 | 3eqtr4ri 2776 | 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 1540 ⊤wtru 1541 ∈ wcel 2108 {csn 4626 〈cop 4632 ∪ cuni 4907 ↦ cmpt 5225 I cid 5577 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 ∈ cmpo 7433 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: txswaphmeo 23813 | 
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