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Theorem seinxp 4497
Description: Intersection of set-like relation with cross 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 4494 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21ancoms 264 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32rabbidva 2607 . . . 4 (𝑥𝐴 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
43eleq1d 2156 . . 3 (𝑥𝐴 → ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
54ralbiia 2392 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6 df-se 4151 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
7 df-se 4151 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
85, 6, 73bitr4i 210 1 (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1438  wral 2359  {crab 2363  Vcvv 2619  cin 2996   class class class wbr 3837   Se wse 4147   × cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-se 4151  df-xp 4434
This theorem is referenced by: (None)
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