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Mirrors > Home > ILE Home > Th. List > df1st2 | GIF version |
Description: An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
df1st2 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 6048 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5342 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | dffn5im 5460 | . . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
4 | 1, 2, 3 | mp2b 8 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
5 | mptv 4020 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
6 | 4, 5 | eqtri 2158 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
7 | 6 | reseq1i 4810 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
8 | resopab 4858 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
9 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | vex 2684 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | op1std 6039 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
12 | 11 | eqeq2d 2149 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
13 | 12 | dfoprab3 6082 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
14 | 7, 8, 13 | 3eqtrri 2163 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 〈cop 3525 {copab 3983 ↦ cmpt 3984 × cxp 4532 ↾ cres 4536 Fn wfn 5113 –onto→wfo 5116 ‘cfv 5118 {coprab 5768 1st c1st 6029 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-oprab 5771 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: (None) |
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