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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 2821 | . 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 2821 | . 2 ⊢ (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fnwelem 7824 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 class class class wbr 5065 {copab 5127 ↦ cmpt 5145 We wwe 5512 × cxp 5552 “ cima 5557 ⟶wf 6350 ‘cfv 6354 1st c1st 7686 2nd c2nd 7687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-1st 7688 df-2nd 7689 |
This theorem is referenced by: r0weon 9437 |
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