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Mirrors > Home > MPE Home > Th. List > df2nd2 | Structured version Visualization version GIF version |
Description: An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
df2nd2 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 8034 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6823 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | dffn5 6967 | . . . . 5 ⊢ (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))) | |
5 | 3, 4 | mpbi 230 | . . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)) |
6 | mptv 5264 | . . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} | |
7 | 5, 6 | eqtri 2763 | . . 3 ⊢ 2nd = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} |
8 | 7 | reseq1i 5996 | . 2 ⊢ (2nd ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) |
9 | resopab 6054 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} | |
10 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
11 | vex 3482 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 10, 11 | op2ndd 8024 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (2nd ‘𝑤) = 𝑦) |
13 | 12 | eqeq2d 2746 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦)) |
14 | 13 | dfoprab3 8078 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} |
15 | 8, 9, 14 | 3eqtrri 2768 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 {copab 5210 ↦ cmpt 5231 × cxp 5687 ↾ cres 5691 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 {coprab 7432 2nd c2nd 8012 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: df2ndres 32720 |
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