Theorem dffr3 6058

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 5585	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		iniseg 6056	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦})
3	2	elv 3445	. . . . . . . 8 ⊢ (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦}
4	3	ineq2i 4169	. . . . . . 7 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
5		dfrab3 4271	. . . . . . 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 3083	. . . 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 1820	. 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 1539   = wceq 1541  {cab 2714   ≠ wne 2932  ∃wrex 3060  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098   Fr wfr 5574  ^◡ccnv 5623   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-fr 5577  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  dffr4  6278  isofrlem  7286  relpfrlem  45190