Theorem dffr3 6129

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 5661	. 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
2		iniseg 6127	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦})
3	2	elv 3493	. . . . . . . 8 ⊢ (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦}
4	3	ineq2i 4238	. . . . . . 7 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
5		dfrab3 4338	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦})
6	4, 5	eqtr4i 2771	. . . . . 6 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦}
7	6	eqeq1i 2745	. . . . 5 ⊢ ((𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)
8	7	rexbii 3100	. . . 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 1817	. 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 1535   = wceq 1537  {cab 2717   ≠ wne 2946  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166   Fr wfr 5649  ^◡ccnv 5699   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-fr 5652  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  dffr4  6352  isofrlem  7376