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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 2736 | . 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 2736 | . 2 ⊢ (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) = (𝑧 ∈ 𝐴 ↦ 〈(𝐹‘𝑧), 𝑧〉) | |
8 | 1, 2, 3, 4, 5, 6, 7 | fnwelem 7876 | 1 ⊢ (𝜑 → 𝑇 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 class class class wbr 5039 {copab 5101 ↦ cmpt 5120 We wwe 5493 × cxp 5534 “ cima 5539 ⟶wf 6354 ‘cfv 6358 1st c1st 7737 2nd c2nd 7738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-1st 7739 df-2nd 7740 |
This theorem is referenced by: r0weon 9591 |
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