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Theorem dffr3 5955
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 5513 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 iniseg 5953 . . . . . . . . 9 (𝑦 ∈ V → (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦})
32elv 3497 . . . . . . . 8 (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦}
43ineq2i 4183 . . . . . . 7 (𝑥 ∩ (𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
5 dfrab3 4275 . . . . . . 7 {𝑧𝑥𝑧𝑅𝑦} = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
64, 5eqtr4i 2844 . . . . . 6 (𝑥 ∩ (𝑅 “ {𝑦})) = {𝑧𝑥𝑧𝑅𝑦}
76eqeq1i 2823 . . . . 5 ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ {𝑧𝑥𝑧𝑅𝑦} = ∅)
87rexbii 3244 . . . 4 (∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
98imbi2i 337 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
109albii 1811 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
111, 10bitr4i 279 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  {cab 2796  wne 3013  wrex 3136  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288  {csn 4557   class class class wbr 5057   Fr wfr 5504  ccnv 5547  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-fr 5507  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  dffr4  6157  isofrlem  7082
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