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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 7080 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6030 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | dffn5 6151 | . . . . 5 ⊢ (2nd Fn V ↔ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))) | |
5 | 3, 4 | mpbi 219 | . . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)) |
6 | mptv 4679 | . . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} | |
7 | 5, 6 | eqtri 2632 | . . 3 ⊢ 2nd = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} |
8 | 7 | reseq1i 5313 | . 2 ⊢ (2nd ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) |
9 | resopab 5366 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} | |
10 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
11 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 10, 11 | op2ndd 7070 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (2nd ‘𝑤) = 𝑦) |
13 | 12 | eqeq2d 2620 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦)) |
14 | 13 | dfoprab3 7115 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} |
15 | 8, 9, 14 | 3eqtrri 2637 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 {copab 4642 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 {coprab 6550 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: df2ndres 28865 |
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