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Theorem txswaphmeolem 22504
 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 id 22 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
21mpompt 7260 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
3 mptresid 5890 . 2 ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧)
4 opelxpi 5561 . . . . . 6 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
54ancoms 462 . . . . 5 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
65adantl 485 . . . 4 ((⊤ ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
7 eqidd 2759 . . . 4 (⊤ → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
8 sneq 4532 . . . . . . . . . 10 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
98cnveqd 5715 . . . . . . . . 9 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
109unieqd 4812 . . . . . . . 8 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
11 opswap 6058 . . . . . . . 8 {⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦
1210, 11eqtrdi 2809 . . . . . . 7 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = ⟨𝑥, 𝑦⟩)
1312mpompt 7260 . . . . . 6 (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1413eqcomi 2767 . . . . 5 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧})
1514a1i 11 . . . 4 (⊤ → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}))
166, 7, 15, 12fmpoco 7795 . . 3 (⊤ → ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩))
1716mptru 1545 . 2 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
182, 3, 173eqtr4ri 2792 1 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {csn 4522  ⟨cop 4528  ∪ cuni 4798   ↦ cmpt 5112   I cid 5429   × cxp 5522  ◡ccnv 5523   ↾ cres 5526   ∘ ccom 5528   ∈ cmpo 7152 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694 This theorem is referenced by:  txswaphmeo  22505
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