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Theorem xrninxp2 34393
Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
Assertion
Ref Expression
xrninxp2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝐶,𝑥   𝑢,𝑅,𝑥   𝑢,𝑆,𝑥

Proof of Theorem xrninxp2
StepHypRef Expression
1 inxp2 34371 . 2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)}
2 3ancoma 1084 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥))
3 df-3an 1074 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥))
4 3anass 1081 . . . 4 ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
52, 3, 43bitr3i 290 . . 3 (((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
65opabbii 4825 . 2 {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
71, 6eqtri 2746 1 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1596  wcel 2103  cin 3679   class class class wbr 4760  {copab 4820   × cxp 5216  cxrn 34214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226
This theorem is referenced by:  inxpxrn  34395
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