Theorem dffr3 6064

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((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (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.

Step	Hyp	Ref	Expression
1		dffr2 5592	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		iniseg 6062	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦})
3	2	elv 3434	. . . . . . . 8 ⊢ (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦}
4	3	ineq2i 4157	. . . . . . 7 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
5		dfrab3 4259	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
6	4, 5	eqtr4i 2762	. . . . . 6 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦}
7	6	eqeq1i 2741	. . . . 5 ⊢ ((𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
8	7	rexbii 3084	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
9	8	imbi2i 336	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
10	9	albii 1821	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
11	1, 10	bitr4i 278	1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  {cab 2714   ≠ wne 2932  ∃wrex 3061  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567   class class class wbr 5085   Fr wfr 5581  ^◡ccnv 5630   “ cima 5634

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-fr 5584  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  dffr4  6284  isofrlem  7295  relpfrlem  45380