Theorem txswaphmeolem 14825

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 6039	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
3		mptresid 5014	. 2 ⊢ ( I ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧)
4		opelxpi 4708	. . . . . 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 2206	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
8		sneq 3644	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
9	8	cnveqd 4855	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ◡{𝑧} = ◡{⟨𝑦, 𝑥⟩})
10	9	unieqd 3861	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑦, 𝑥⟩})
11		vex 2775	. . . . . . . . 9 ⊢ 𝑦 ∈ V
12		vex 2775	. . . . . . . . 9 ⊢ 𝑥 ∈ V
13		opswapg 5170	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩)
14	11, 12, 13	mp2an 426	. . . . . . . 8 ⊢ ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩
15	10, 14	eqtrdi 2254	. . . . . . 7 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ⟨𝑥, 𝑦⟩)
16	15	mpompt 6039	. . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
17	16	eqcomi 2209	. . . . 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 6304	. . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩))
20	19	mptru 1382	. 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
21	2, 3, 20	3eqtr4ri 2237	1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Syntax hints:   ∧ wa 104   = wceq 1373  ⊤wtru 1374   ∈ wcel 2176  Vcvv 2772  {csn 3633  ⟨cop 3636  ∪ cuni 3850   ↦ cmpt 4106   I cid 4336   × cxp 4674  ^◡ccnv 4675   ↾ cres 4678   ∘ ccom 4680   ∈ cmpo 5948

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229

This theorem is referenced by:  txswaphmeo  14826