Theorem txswaphmeolem 22955

Description: Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
txswaphmeolem	⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
2	1	mpompt 7388	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
3		mptresid 5958	. 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧)
4		opelxpi 5626	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
5	4	ancoms 459	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
6	5	adantl 482	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
7		eqidd 2739	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
8		sneq 4571	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
9	8	cnveqd 5784	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ◡{𝑧} = ◡{⟨𝑦, 𝑥⟩})
10	9	unieqd 4853	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑦, 𝑥⟩})
11		opswap 6132	. . . . . . . 8 ⊢ ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩
12	10, 11	eqtrdi 2794	. . . . . . 7 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ⟨𝑥, 𝑦⟩)
13	12	mpompt 7388	. . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
14	13	eqcomi 2747	. . . . 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 7935	. . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩))
17	16	mptru 1546	. 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
18	2, 3, 17	3eqtr4ri 2777	1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Syntax hints:   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  {csn 4561  ⟨cop 4567  ∪ cuni 4839   ↦ cmpt 5157   I cid 5488   × cxp 5587  ^◡ccnv 5588   ↾ cres 5591   ∘ ccom 5593   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  txswaphmeo  22956