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Theorem txswaphmeolem 12720
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 4618 . . . . . 6 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
21ancoms 266 . . . . 5 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
32adantl 275 . . . 4 ((⊤ ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
4 eqidd 2158 . . . 4 (⊤ → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))
5 sneq 3571 . . . . . . . . . 10 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
65cnveqd 4762 . . . . . . . . 9 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
76unieqd 3783 . . . . . . . 8 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
8 vex 2715 . . . . . . . . 9 𝑦 ∈ V
9 vex 2715 . . . . . . . . 9 𝑥 ∈ V
10 opswapg 5072 . . . . . . . . 9 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩)
118, 9, 10mp2an 423 . . . . . . . 8 {⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦
127, 11eqtrdi 2206 . . . . . . 7 (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = ⟨𝑥, 𝑦⟩)
1312mpompt 5913 . . . . . 6 (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1413eqcomi 2161 . . . . 5 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧})
1514a1i 9 . . . 4 (⊤ → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ {𝑧}))
163, 4, 15, 12fmpoco 6163 . . 3 (⊤ → ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩))
1716mptru 1344 . 2 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
18 id 19 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
1918mpompt 5913 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑥, 𝑦⟩)
20 mptresid 4920 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌))
2117, 19, 203eqtr2i 2184 1 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wtru 1336  wcel 2128  Vcvv 2712  {csn 3560  cop 3563   cuni 3772  cmpt 4025   I cid 4248   × cxp 4584  ccnv 4585  cres 4588  ccom 4590  cmpo 5826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089
This theorem is referenced by:  txswaphmeo  12721
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