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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 6105 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5394 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | dffn5im 5514 | . . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
4 | 1, 2, 3 | mp2b 8 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
5 | mptv 4061 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
6 | 4, 5 | eqtri 2178 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
7 | 6 | reseq1i 4862 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
8 | resopab 4910 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
9 | vex 2715 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | vex 2715 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | op1std 6096 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
12 | 11 | eqeq2d 2169 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
13 | 12 | dfoprab3 6139 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
14 | 7, 8, 13 | 3eqtrri 2183 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∈ wcel 2128 Vcvv 2712 〈cop 3563 {copab 4024 ↦ cmpt 4025 × cxp 4584 ↾ cres 4588 Fn wfn 5165 –onto→wfo 5168 ‘cfv 5170 {coprab 5825 1st c1st 6086 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 df-oprab 5828 df-1st 6088 df-2nd 6089 |
This theorem is referenced by: (None) |
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