Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7268 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑥, 𝑦〉) |
3 | mptresid 5920 | . 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) | |
4 | opelxpi 5594 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) | |
5 | 4 | ancoms 461 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
6 | 5 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 〈𝑦, 𝑥〉 ∈ (𝑌 × 𝑋)) |
7 | eqidd 2824 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) | |
8 | sneq 4579 | . . . . . . . . . 10 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → {𝑧} = {〈𝑦, 𝑥〉}) | |
9 | 8 | cnveqd 5748 | . . . . . . . . 9 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ◡{𝑧} = ◡{〈𝑦, 𝑥〉}) |
10 | 9 | unieqd 4854 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = ∪ ◡{〈𝑦, 𝑥〉}) |
11 | opswap 6088 | . . . . . . . 8 ⊢ ∪ ◡{〈𝑦, 𝑥〉} = 〈𝑥, 𝑦〉 | |
12 | 10, 11 | syl6eq 2874 | . . . . . . 7 ⊢ (𝑧 = 〈𝑦, 𝑥〉 → ∪ ◡{𝑧} = 〈𝑥, 𝑦〉) |
13 | 12 | mpompt 7268 | . . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) |
14 | 13 | eqcomi 2832 | . . . . 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 7792 | . . 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 2857 | 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 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 {csn 4569 〈cop 4575 ∪ cuni 4840 ↦ cmpt 5148 I cid 5461 × cxp 5555 ◡ccnv 5556 ↾ cres 5559 ∘ ccom 5561 ∈ cmpo 7160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: txswaphmeo 22415 |
Copyright terms: Public domain | W3C validator |