Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5498 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) | |
2 | df-pred 5680 | . . . . . 6 ⊢ Pred(𝑅, 𝑥, 𝑦) = (𝑥 ∩ (◡𝑅 “ {𝑦})) | |
3 | 2 | eqeq1i 2627 | . . . . 5 ⊢ (Pred(𝑅, 𝑥, 𝑦) = ∅ ↔ (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) |
4 | 3 | rexbii 3041 | . . . 4 ⊢ (∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅) |
5 | 4 | imbi2i 326 | . . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
6 | 5 | albii 1747 | . 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ (◡𝑅 “ {𝑦})) = ∅)) |
7 | 1, 6 | bitr4i 267 | 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 196 ∧ wa 384 ∀wal 1481 = wceq 1483 ≠ wne 2794 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 Fr wfr 5070 ◡ccnv 5113 “ cima 5117 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-fr 5073 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: frmin 31739 |
Copyright terms: Public domain | W3C validator |