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Theorem frinxp 5759
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 3976 . . . . . . . . . . 11 (𝑧𝐴 → (𝑥𝑧𝑥𝐴))
2 ssel 3976 . . . . . . . . . . 11 (𝑧𝐴 → (𝑦𝑧𝑦𝐴))
31, 2anim12d 610 . . . . . . . . . 10 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑥𝐴𝑦𝐴)))
4 brinxp 5755 . . . . . . . . . . 11 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 460 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
63, 5syl6 35 . . . . . . . . 9 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76impl 457 . . . . . . . 8 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
87notbid 318 . . . . . . 7 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
98ralbidva 3176 . . . . . 6 ((𝑧𝐴𝑥𝑧) → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
109rexbidva 3177 . . . . 5 (𝑧𝐴 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1110adantr 482 . . . 4 ((𝑧𝐴𝑧 ≠ ∅) → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1211pm5.74i 271 . . 3 (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1312albii 1822 . 2 (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
14 df-fr 5632 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
15 df-fr 5632 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1613, 14, 153bitr4i 303 1 (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wcel 2107  wne 2941  wral 3062  wrex 3071  cin 3948  wss 3949  c0 4323   class class class wbr 5149   Fr wfr 5629   × cxp 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-fr 5632  df-xp 5683
This theorem is referenced by:  weinxp  5761
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