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Theorem frinxp 5768
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 3977 . . . . . . . . . . 11 (𝑧𝐴 → (𝑥𝑧𝑥𝐴))
2 ssel 3977 . . . . . . . . . . 11 (𝑧𝐴 → (𝑦𝑧𝑦𝐴))
31, 2anim12d 609 . . . . . . . . . 10 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑥𝐴𝑦𝐴)))
4 brinxp 5764 . . . . . . . . . . 11 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 458 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
63, 5syl6 35 . . . . . . . . 9 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76impl 455 . . . . . . . 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 480 . . . 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 1819 . 2 (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
14 df-fr 5637 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
15 df-fr 5637 . 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 206  wa 395  wal 1538  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333   class class class wbr 5143   Fr wfr 5634   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-fr 5637  df-xp 5691
This theorem is referenced by:  weinxp  5770
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