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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 8034 | . . . . . 6 ⊢ 1st :V–onto→V | |
| 2 | fofn 6822 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
| 3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
| 4 | dffn5 6967 | . . . . 5 ⊢ (1st Fn V ↔ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
| 5 | 3, 4 | mpbi 230 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
| 6 | mptv 5258 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
| 7 | 5, 6 | eqtri 2765 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
| 8 | 7 | reseq1i 5993 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
| 9 | resopab 6052 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
| 10 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 11 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 12 | 10, 11 | op1std 8024 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
| 13 | 12 | eqeq2d 2748 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
| 14 | 13 | dfoprab3 8079 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
| 15 | 8, 9, 14 | 3eqtrri 2770 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {copab 5205 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 {coprab 7432 1st c1st 8012 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-oprab 7435 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: df1stres 32713 |
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