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Mirrors > Home > MPE Home > Th. List > dffr3 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dffr3 | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffr2 5003 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | |
2 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
3 | iniseg 5415 | . . . . . . . . 9 ⊢ (𝑦 ∈ V → (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦}) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ (◡𝑅 “ {𝑦}) = {𝑧 ∣ 𝑧𝑅𝑦} |
5 | 4 | ineq2i 3773 | . . . . . . 7 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦}) |
6 | dfrab3 3861 | . . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = (𝑥 ∩ {𝑧 ∣ 𝑧𝑅𝑦}) | |
7 | 5, 6 | eqtr4i 2635 | . . . . . 6 ⊢ (𝑥 ∩ (◡𝑅 “ {𝑦})) = {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} |
8 | 7 | eqeq1i 2615 | . . . . 5 ⊢ ((𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅) |
9 | 8 | rexbii 3023 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅ ↔ ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅) |
10 | 9 | imbi2i 325 | . . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) |
11 | 10 | albii 1737 | . 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) |
12 | 1, 11 | bitr4i 266 | 1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 Fr wfr 4994 ◡ccnv 5037 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-fr 4997 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: dffr4 5613 isofrlem 6490 |
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