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| Mirrors > Home > MPE Home > Th. List > fnwe | Structured version Visualization version GIF version | ||
| Description: A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
| Ref | Expression |
|---|---|
| fnwe.1 | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} |
| fnwe.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| fnwe.3 | ⊢ (𝜑 → 𝑅 We 𝐵) |
| fnwe.4 | ⊢ (𝜑 → 𝑆 We 𝐴) |
| fnwe.5 | ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V) |
| Ref | Expression |
|---|---|
| fnwe | ⊢ (𝜑 → 𝑇 We 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnwe.1 | . 2 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} | |
| 2 | fnwe.2 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 3 | fnwe.3 | . 2 ⊢ (𝜑 → 𝑅 We 𝐵) | |
| 4 | fnwe.4 | . 2 ⊢ (𝜑 → 𝑆 We 𝐴) | |
| 5 | fnwe.5 | . 2 ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V) | |
| 6 | eqid 2737 | . 2 ⊢ {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))} = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))} | |
| 7 | eqid 2737 | . 2 ⊢ (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | fnwelem 8156 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 We wwe 5636 × cxp 5683 “ cima 5688 ⟶wf 6557 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: r0weon 10052 |
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