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Mirrors > Home > ILE Home > Th. List > df2nd2 | 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 6024 | . . . . 5 ⊢ 2nd :V–onto→V | |
2 | fofn 5317 | . . . . 5 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | dffn5im 5435 | . . . . 5 ⊢ (2nd Fn V → 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤))) | |
4 | 1, 2, 3 | mp2b 8 | . . . 4 ⊢ 2nd = (𝑤 ∈ V ↦ (2nd ‘𝑤)) |
5 | mptv 3995 | . . . 4 ⊢ (𝑤 ∈ V ↦ (2nd ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} | |
6 | 4, 5 | eqtri 2138 | . . 3 ⊢ 2nd = {〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} |
7 | 6 | reseq1i 4785 | . 2 ⊢ (2nd ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) |
8 | resopab 4833 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (2nd ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} | |
9 | vex 2663 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | vex 2663 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | op2ndd 6015 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (2nd ‘𝑤) = 𝑦) |
12 | 11 | eqeq2d 2129 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (2nd ‘𝑤) ↔ 𝑧 = 𝑦)) |
13 | 12 | dfoprab3 6057 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (2nd ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} |
14 | 7, 8, 13 | 3eqtrri 2143 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 〈cop 3500 {copab 3958 ↦ cmpt 3959 × cxp 4507 ↾ cres 4511 Fn wfn 5088 –onto→wfo 5091 ‘cfv 5093 {coprab 5743 2nd c2nd 6005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-oprab 5746 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: (None) |
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