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Theorem frinxp 5750
Description: Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
frinxp (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Proof of Theorem frinxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3971 . . . . . . . . . . 11 (𝑧𝐴 → (𝑥𝑧𝑥𝐴))
2 ssel 3971 . . . . . . . . . . 11 (𝑧𝐴 → (𝑦𝑧𝑦𝐴))
31, 2anim12d 609 . . . . . . . . . 10 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑥𝐴𝑦𝐴)))
4 brinxp 5746 . . . . . . . . . . 11 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 459 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
63, 5syl6 35 . . . . . . . . 9 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76impl 456 . . . . . . . 8 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
87notbid 317 . . . . . . 7 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
98ralbidva 3174 . . . . . 6 ((𝑧𝐴𝑥𝑧) → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
109rexbidva 3175 . . . . 5 (𝑧𝐴 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1110adantr 481 . . . 4 ((𝑧𝐴𝑧 ≠ ∅) → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1211pm5.74i 270 . . 3 (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1312albii 1821 . 2 (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
14 df-fr 5624 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
15 df-fr 5624 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1613, 14, 153bitr4i 302 1 (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539  wcel 2106  wne 2939  wral 3060  wrex 3069  cin 3943  wss 3944  c0 4318   class class class wbr 5141   Fr wfr 5621   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-fr 5624  df-xp 5675
This theorem is referenced by:  weinxp  5752
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