Theorem xrninxp 34469

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 34448	. . 3 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)}
2		df-3an 1074	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))
3		3anan12 1082	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
4	2, 3	bitr3i 266	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
5	4	opabbii 4865	. . 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 2778	. 2 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
7		cnvopab 5687	. 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 4804	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))
9	8	anbi2d 742	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩)))
10	9	dfoprab4 7388	. . 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 5448	. 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 2784	1 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}

Syntax hints:   ∧ wa 383   ∧ w3a 1072   = wceq 1628   ∈ wcel 2135   ∩ cin 3710  ⟨cop 4323   class class class wbr 4800  {copab 4860   × cxp 5260  ^◡ccnv 5261  {coprab 6810   ⋉ cxrn 34291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-iota 6008  df-fun 6047  df-fv 6053  df-oprab 6813  df-1st 7329  df-2nd 7330

This theorem is referenced by: (None)