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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 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨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.
StepHypRef Expression
1 inxp2 34448 . . 3 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)}
2 df-3an 1074 . . . . 5 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥))
3 3anan12 1082 . . . . 5 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
42, 3bitr3i 266 . . . 4 (((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
54opabbii 4865 . . 3 {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
61, 5eqtri 2778 . 2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
7 cnvopab 5687 . 2 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
8 breq2 4804 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑢(𝑅𝑆)𝑥𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))
98anbi2d 742 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩)))
109dfoprab4 7388 . . 3 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))} = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))}
1110cnveqi 5448 . 2 {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))} = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))}
126, 7, 113eqtr2i 2784 1 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)⟨𝑦, 𝑧⟩))}
Colors of variables: wff setvar class
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)
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