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Theorem dffr3 5417
 Description: Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dffr3 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dffr3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffr2 5003 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 vex 3176 . . . . . . . . 9 𝑦 ∈ V
3 iniseg 5415 . . . . . . . . 9 (𝑦 ∈ V → (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦})
42, 3ax-mp 5 . . . . . . . 8 (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦}
54ineq2i 3773 . . . . . . 7 (𝑥 ∩ (𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
6 dfrab3 3861 . . . . . . 7 {𝑧𝑥𝑧𝑅𝑦} = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
75, 6eqtr4i 2635 . . . . . 6 (𝑥 ∩ (𝑅 “ {𝑦})) = {𝑧𝑥𝑧𝑅𝑦}
87eqeq1i 2615 . . . . 5 ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ {𝑧𝑥𝑧𝑅𝑦} = ∅)
98rexbii 3023 . . . 4 (∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
109imbi2i 325 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
1110albii 1737 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
121, 11bitr4i 266 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583   Fr wfr 4994  ◡ccnv 5037   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-fr 4997  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  dffr4  5613  isofrlem  6490
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