Theorem xrninxp 36445

Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.)

Assertion

Ref	Expression
xrninxp	⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}

Distinct variable groups:   𝑢,𝐴,𝑦,𝑧   𝑢,𝐵,𝑦,𝑧   𝑢,𝐶,𝑦,𝑧   𝑢,𝑅,𝑦,𝑧   𝑢,𝑆,𝑦,𝑧

Proof of Theorem xrninxp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inxp2 36424	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)}
2		df-3an 1087	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))
3		3anan12 1094	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
4	2, 3	bitr3i 276	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
5	4	opabbii 5137	. . 3 ⊢ {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
6	1, 5	eqtri 2766	. 2 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
7		cnvopab 6031	. 2 ⊢ ◡{⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
8		breq2 5074	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))
9	8	anbi2d 628	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩)))
10	9	dfoprab4 7868	. . 3 ⊢ {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}
11	10	cnveqi 5772	. 2 ⊢ ◡{⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}
12	6, 7, 11	3eqtr2i 2772	1 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}

Syntax hints:   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∩ cin 3882  ⟨cop 4564   class class class wbr 5070  {copab 5132   × cxp 5578  ^◡ccnv 5579  {coprab 7256   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-oprab 7259  df-1st 7804  df-2nd 7805

This theorem is referenced by: (None)