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Theorem txswaphmeolem 21417
 Description: Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
txswaphmeolem ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦

Proof of Theorem txswaphmeolem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 opelxpi 5072 . . . . . 6 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
21ancoms 468 . . . . 5 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
32adantl 481 . . . 4 ((⊤ ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
4 eqidd 2611 . . . 4 (⊤ → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
5 sneq 4135 . . . . . . . . . 10 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
65cnveqd 5220 . . . . . . . . 9 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
76unieqd 4382 . . . . . . . 8 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
8 opswap 5540 . . . . . . . 8 {⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦
97, 8syl6eq 2660 . . . . . . 7 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = ⟨𝑥, 𝑦⟩)
109mpt2mpt 6650 . . . . . 6 (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1110eqcomi 2619 . . . . 5 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧})
1211a1i 11 . . . 4 (⊤ → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}))
133, 4, 12, 9fmpt2co 7147 . . 3 (⊤ → ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩))
1413trud 1484 . 2 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
15 id 22 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
1615mpt2mpt 6650 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
17 mptresid 5375 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌))
1814, 16, 173eqtr2i 2638 1 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨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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