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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 2734 | . 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 2734 | . 2 ⊢ (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fnwelem 8154 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 class class class wbr 5147 {copab 5209 ↦ cmpt 5230 We wwe 5639 × cxp 5686 “ cima 5691 ⟶wf 6558 ‘cfv 6562 1st c1st 8010 2nd c2nd 8011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-1st 8012 df-2nd 8013 |
This theorem is referenced by: r0weon 10049 |
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