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Theorem dffr3 5838
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 5408 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
2 iniseg 5836 . . . . . . . . 9 (𝑦 ∈ V → (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦})
32elv 3442 . . . . . . . 8 (𝑅 “ {𝑦}) = {𝑧𝑧𝑅𝑦}
43ineq2i 4106 . . . . . . 7 (𝑥 ∩ (𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
5 dfrab3 4198 . . . . . . 7 {𝑧𝑥𝑧𝑅𝑦} = (𝑥 ∩ {𝑧𝑧𝑅𝑦})
64, 5eqtr4i 2822 . . . . . 6 (𝑥 ∩ (𝑅 “ {𝑦})) = {𝑧𝑥𝑧𝑅𝑦}
76eqeq1i 2800 . . . . 5 ((𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ {𝑧𝑥𝑧𝑅𝑦} = ∅)
87rexbii 3211 . . . 4 (∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅)
98imbi2i 337 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 (𝑥 ∩ (𝑅 “ {𝑦})) = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
109albii 1801 . 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 1520   = wceq 1522  {cab 2775  wne 2984  wrex 3106  {crab 3109  Vcvv 3437  cin 3858  wss 3859  c0 4211  {csn 4472   class class class wbr 4962   Fr wfr 5399  ccnv 5442  cima 5446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-fr 5402  df-xp 5449  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456
This theorem is referenced by:  dffr4  6039  isofrlem  6956
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