Theorem txswaphmeolem 13905

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		opelxpi 4660	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
2	1	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
3	2	adantl 277	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
4		eqidd 2178	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
5		sneq 3605	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → {𝑧} = {⟨𝑦, 𝑥⟩})
6	5	cnveqd 4805	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ◡{𝑧} = ◡{⟨𝑦, 𝑥⟩})
7	6	unieqd 3822	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ∪ ◡{⟨𝑦, 𝑥⟩})
8		vex 2742	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9		vex 2742	. . . . . . . . 9 ⊢ 𝑥 ∈ V
10		opswapg 5117	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩)
11	8, 9, 10	mp2an 426	. . . . . . . 8 ⊢ ∪ ◡{⟨𝑦, 𝑥⟩} = ⟨𝑥, 𝑦⟩
12	7, 11	eqtrdi 2226	. . . . . . 7 ⊢ (𝑧 = ⟨𝑦, 𝑥⟩ → ∪ ◡{𝑧} = ⟨𝑥, 𝑦⟩)
13	12	mpompt 5969	. . . . . 6 ⊢ (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
14	13	eqcomi 2181	. . . . 5 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧})
15	14	a1i 9	. . . 4 ⊢ (⊤ → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ∪ ◡{𝑧}))
16	3, 4, 15, 12	fmpoco 6219	. . 3 ⊢ (⊤ → ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩))
17	16	mptru 1362	. 2 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
18		id 19	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑥, 𝑦⟩)
19	18	mpompt 5969	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑥, 𝑦⟩)
20		mptresid 4963	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑧) = ( I ↾ (𝑋 × 𝑌))
21	17, 19, 20	3eqtr2i 2204	1 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))

Syntax hints:   ∧ wa 104   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148  Vcvv 2739  {csn 3594  ⟨cop 3597  ∪ cuni 3811   ↦ cmpt 4066   I cid 4290   × cxp 4626  ^◡ccnv 4627   ↾ cres 4630   ∘ ccom 4632   ∈ cmpo 5879

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144

This theorem is referenced by:  txswaphmeo  13906