Theorem dffr3 2620

Description: Alternate definition of founded relation. Definition 6.21 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
dffr3	⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ (◡R “ {y})) = ∅))

Distinct variable group(s):   x,y,R   x,A

Proof of Theorem dffr3

Step	Hyp	Ref	Expression
1		dffr2 2171	. 2 ⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))
2		visset 1350	. . . . . . . 8 ⊢ y ∈ V
3		iniseg 2619	. . . . . . . 8 ⊢ (y ∈ V → (◡R “ {y}) = {z∣zRy})
4	2, 3	ax-mp 6	. . . . . . 7 ⊢ (◡R “ {y}) = {z∣zRy}
5	4	ineq2i 1642	. . . . . 6 ⊢ (x ∩ (◡R “ {y})) = (x ∩ {z∣zRy})
6	5	cleq1i 1108	. . . . 5 ⊢ ((x ∩ (◡R “ {y})) = ∅ ↔ (x ∩ {z∣zRy}) = ∅)
7	6	birex 1224	. . . 4 ⊢ (∃y ∈ x (x ∩ (◡R “ {y})) = ∅ ↔ ∃y ∈ x (x ∩ {z∣zRy}) = ∅)
8	7	imbi2i 160	. . 3 ⊢ (((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ (◡R “ {y})) = ∅) ↔ ((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))
9	8	bial 695	. 2 ⊢ (∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ (◡R “ {y})) = ∅) ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))
10	1, 9	bitr4 154	1 ⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ (◡R “ {y})) = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054   Fr wfr 2061  ^◡ccnv 2409   “ cima 2413

This theorem is referenced by:  isofrlem 2939

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-fr 2169  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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