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Theorem seinxp 5097
Description: Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
seinxp (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Proof of Theorem seinxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brinxp 5093 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21ancoms 467 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32rabbidva 3162 . . . 4 (𝑥𝐴 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
43eleq1d 2671 . . 3 (𝑥𝐴 → ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
54ralbiia 2961 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6 df-se 4987 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
7 df-se 4987 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
85, 6, 73bitr4i 290 1 (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cin 3538   class class class wbr 4577   Se wse 4984   × cxp 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-se 4987  df-xp 5033
This theorem is referenced by: (None)
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