Theorem df1stres 32686

Description: Definition for a restriction of the 1^st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Assertion

Ref	Expression
df1stres	⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem df1stres

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1st2 8102	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
2	1	reseq1i 5967	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵))
3		resoprab 7530	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
4		resres 5984	. . . 4 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5		incom 4189	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6		xpss 5675	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
7		dfss2 3949	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
8	6, 7	mpbi 230	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
9	5, 8	eqtr3i 2761	. . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
10	9	reseq2i 5968	. . . 4 ⊢ (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × 𝐵))
11	4, 10	eqtri 2759	. . 3 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ (𝐴 × 𝐵))
12	2, 3, 11	3eqtr3ri 2768	. 2 ⊢ (1st ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
13		df-mpo 7415	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
14	12, 13	eqtr4i 2762	1 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931   × cxp 5657   ↾ cres 5661  {coprab 7411   ∈ cmpo 7412  1^st c1st 7991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994

This theorem is referenced by:  cnre2csqima  33947