Theorem df1stres 32627

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 8077	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
2	1	reseq1i 5946	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵))
3		resoprab 7507	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
4		resres 5963	. . . 4 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5		incom 4172	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6		xpss 5654	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
7		dfss2 3932	. . . . . . 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 2754	. . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
10	9	reseq2i 5947	. . . 4 ⊢ (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × 𝐵))
11	4, 10	eqtri 2752	. . 3 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ (𝐴 × 𝐵))
12	2, 3, 11	3eqtr3ri 2761	. 2 ⊢ (1st ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
13		df-mpo 7392	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)}
14	12, 13	eqtr4i 2755	1 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914   × cxp 5636   ↾ cres 5640  {coprab 7388   ∈ cmpo 7389  1^st c1st 7966

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969

This theorem is referenced by:  cnre2csqima  33901