Theorem txswaphmeolem 14499

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 19	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
2	1	mpompt 6011	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
3		mptresid 4997	. 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧)
4		opelxpi 4692	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
5	4	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
6	5	adantl 277	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
7		eqidd 2194	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
8		sneq 3630	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
9	8	cnveqd 4839	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ◡{𝑧} = ◡{⟨𝑦, 𝑥⟩})
10	9	unieqd 3847	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑦, 𝑥⟩})
11		vex 2763	. . . . . . . . 9 ⊢ 𝑦 ∈ V
12		vex 2763	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13		opswapg 5153	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩)
14	11, 12, 13	mp2an 426	. . . . . . . 8 ⊢ ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩
15	10, 14	eqtrdi 2242	. . . . . . 7 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ⟨𝑥, 𝑦⟩)
16	15	mpompt 6011	. . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
17	16	eqcomi 2197	. . . . 5 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧})
18	17	a1i 9	. . . 4 ⊢ (⊤ → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}))
19	6, 7, 18, 15	fmpoco 6271	. . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩))
20	19	mptru 1373	. 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
21	2, 3, 20	3eqtr4ri 2225	1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Syntax hints:   ∧ wa 104   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  Vcvv 2760  {csn 3619  ⟨cop 3622  ∪ cuni 3836   ↦ cmpt 4091   I cid 4320   × cxp 4658  ^◡ccnv 4659   ↾ cres 4662   ∘ ccom 4664   ∈ cmpo 5921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196

This theorem is referenced by:  txswaphmeo  14500