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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 2777 | . 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 2777 | . 2 ⊢ (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fnwelem 7573 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2106 Vcvv 3397 〈cop 4403 class class class wbr 4886 {copab 4948 ↦ cmpt 4965 We wwe 5313 × cxp 5353 “ cima 5358 ⟶wf 6131 ‘cfv 6135 1st c1st 7443 2nd c2nd 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-int 4711 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-1st 7445 df-2nd 7446 |
This theorem is referenced by: r0weon 9168 |
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