Theorem dffr3 6088

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 5630	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		iniseg 6086	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦})
3	2	elv 3472	. . . . . . . 8 ⊢ (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦}
4	3	ineq2i 4201	. . . . . . 7 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
5		dfrab3 4301	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
6	4, 5	eqtr4i 2755	. . . . . 6 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦}
7	6	eqeq1i 2729	. . . . 5 ⊢ ((𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
8	7	rexbii 3086	. . . 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 1813	. 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 205   ∧ wa 395  ∀wal 1531   = wceq 1533  {cab 2701   ≠ wne 2932  ∃wrex 3062  {crab 3424  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314  {csn 4620   class class class wbr 5138   Fr wfr 5618  ^◡ccnv 5665   “ cima 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-fr 5621  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is referenced by:  dffr4  6310  isofrlem  7329