Theorem xrninxp 38393

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 38368	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)}
2		df-3an 1089	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))
3		3anan12 1096	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
4	2, 3	bitr3i 277	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
5	4	opabbii 5210	. . 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 2765	. 2 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
7		cnvopab 6157	. 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 5147	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))
9	8	anbi2d 630	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩)))
10	9	dfoprab4 8080	. . 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 5885	. 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 2771	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 1087   = wceq 1540   ∈ wcel 2108   ∩ cin 3950  ⟨cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  ^◡ccnv 5684  {coprab 7432   ⋉ cxrn 38181

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-1st 8014  df-2nd 8015

This theorem is referenced by: (None)