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Mirrors > Home > MPE Home > Th. List > dffr4 | Structured version Visualization version GIF version |
Description: Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
dffr4 | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffr3 6051 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) | |
2 | df-pred 6253 | . . . . . 6 ⊢ Pred(𝑅, 𝑥, 𝑦) = (𝑥 ∩ (◡𝑅 “ {𝑦})) | |
3 | 2 | eqeq1i 2741 | . . . . 5 ⊢ (Pred(𝑅, 𝑥, 𝑦) = ∅ ↔ (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) |
4 | 3 | rexbii 3097 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) |
5 | 4 | imbi2i 335 | . . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
6 | 5 | albii 1821 | . 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
7 | 1, 6 | bitr4i 277 | 1 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ≠ wne 2943 ∃wrex 3073 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 {csn 4586 Fr wfr 5585 ◡ccnv 5632 “ cima 5636 Predcpred 6252 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-fr 5588 df-xp 5639 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 |
This theorem is referenced by: frmin 9684 |
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