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Theorem dffr3 5934
 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 5489 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 iniseg 5932 . . . . . . . . 9 (𝑦 ∈ V → (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦})
32elv 3415 . . . . . . . 8 (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦}
43ineq2i 4114 . . . . . . 7 (𝑥 ∩ (𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
5 dfrab3 4212 . . . . . . 7 {𝑧𝑥𝑧𝑅𝑦} = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
64, 5eqtr4i 2784 . . . . . 6 (𝑥 ∩ (𝑅 “ {𝑦})) = {𝑧𝑥𝑧𝑅𝑦}
76eqeq1i 2763 . . . . 5 ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ {𝑧𝑥𝑧𝑅𝑦} = ∅)
87rexbii 3175 . . . 4 (∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
98imbi2i 339 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
109albii 1821 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
111, 10bitr4i 281 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  {cab 2735   ≠ wne 2951  ∃wrex 3071  {crab 3074  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {csn 4522   class class class wbr 5032   Fr wfr 5480  ◡ccnv 5523   “ cima 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-fr 5483  df-xp 5530  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537 This theorem is referenced by:  dffr4  6142  isofrlem  7087
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