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Mirrors > Home > MPE Home > Th. List > df1st2 | Structured version Visualization version 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 8014 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6812 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | dffn5 6956 | . . . . 5 ⊢ (1st Fn V ↔ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
5 | 3, 4 | mpbi 229 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
6 | mptv 5265 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
7 | 5, 6 | eqtri 2753 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
8 | 7 | reseq1i 5981 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
9 | resopab 6039 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
10 | vex 3465 | . . . . 5 ⊢ 𝑥 ∈ V | |
11 | vex 3465 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 10, 11 | op1std 8004 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
13 | 12 | eqeq2d 2736 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
14 | 13 | dfoprab3 8059 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
15 | 8, 9, 14 | 3eqtrri 2758 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 〈cop 4636 {copab 5211 ↦ cmpt 5232 × cxp 5676 ↾ cres 5680 Fn wfn 6544 –onto→wfo 6547 ‘cfv 6549 {coprab 7420 1st c1st 7992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fo 6555 df-fv 6557 df-oprab 7423 df-1st 7994 df-2nd 7995 |
This theorem is referenced by: df1stres 32565 |
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