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Theorem dffr3 6088
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 5630 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 iniseg 6086 . . . . . . . . 9 (𝑦 ∈ V → (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦})
32elv 3472 . . . . . . . 8 (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦}
43ineq2i 4201 . . . . . . 7 (𝑥 ∩ (𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
5 dfrab3 4301 . . . . . . 7 {𝑧𝑥𝑧𝑅𝑦} = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
64, 5eqtr4i 2755 . . . . . 6 (𝑥 ∩ (𝑅 “ {𝑦})) = {𝑧𝑥𝑧𝑅𝑦}
76eqeq1i 2729 . . . . 5 ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ {𝑧𝑥𝑧𝑅𝑦} = ∅)
87rexbii 3086 . . . 4 (∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
98imbi2i 336 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
109albii 1813 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
111, 10bitr4i 278 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  {cab 2701  wne 2932  wrex 3062  {crab 3424  Vcvv 3466  cin 3939  wss 3940  c0 4314  {csn 4620   class class class wbr 5138   Fr wfr 5618  ccnv 5665  cima 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-fr 5621  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679
This theorem is referenced by:  dffr4  6310  isofrlem  7329
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