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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 6063 | . . . . 5 ⊢ 1st :V–onto→V | |
2 | fofn 5355 | . . . . 5 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | dffn5im 5475 | . . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
4 | 1, 2, 3 | mp2b 8 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
5 | mptv 4033 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
6 | 4, 5 | eqtri 2161 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
7 | 6 | reseq1i 4823 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
8 | resopab 4871 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
9 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
10 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | op1std 6054 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
12 | 11 | eqeq2d 2152 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
13 | 12 | dfoprab3 6097 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
14 | 7, 8, 13 | 3eqtrri 2166 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 {copab 3996 ↦ cmpt 3997 × cxp 4545 ↾ cres 4549 Fn wfn 5126 –onto→wfo 5129 ‘cfv 5131 {coprab 5783 1st c1st 6044 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-oprab 5786 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: (None) |
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