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Theorem txswaphmeolem 21512
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 5113 . . . . . 6 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
21ancoms 469 . . . . 5 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
32adantl 482 . . . 4 ((⊤ ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
4 eqidd 2627 . . . 4 (⊤ → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
5 sneq 4163 . . . . . . . . . 10 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
65cnveqd 5263 . . . . . . . . 9 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
76unieqd 4417 . . . . . . . 8 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
8 opswap 5584 . . . . . . . 8 {⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦
97, 8syl6eq 2676 . . . . . . 7 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = ⟨𝑥, 𝑦⟩)
109mpt2mpt 6706 . . . . . 6 (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1110eqcomi 2635 . . . . 5 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧})
1211a1i 11 . . . 4 (⊤ → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}))
133, 4, 12, 9fmpt2co 7206 . . 3 (⊤ → ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩))
1413trud 1490 . 2 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
15 id 22 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
1615mpt2mpt 6706 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
17 mptresid 5419 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌))
1814, 16, 173eqtr2i 2654 1 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wtru 1481  wcel 1992  {csn 4153  cop 4159   cuni 4407  cmpt 4678   I cid 4989   × cxp 5077  ccnv 5078  cres 5081  ccom 5083  cmpt2 6607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117
This theorem is referenced by:  txswaphmeo  21513
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