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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 7188 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6117 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | dffn5 6241 | . . . . 5 ⊢ (1st Fn V ↔ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))) | |
5 | 3, 4 | mpbi 220 | . . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)) |
6 | mptv 4751 | . . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} | |
7 | 5, 6 | eqtri 2644 | . . 3 ⊢ 1st = {〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} |
8 | 7 | reseq1i 5392 | . 2 ⊢ (1st ↾ (V × V)) = ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) |
9 | resopab 5446 | . 2 ⊢ ({〈𝑤, 𝑧〉 ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} | |
10 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
11 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 10, 11 | op1std 7178 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (1st ‘𝑤) = 𝑥) |
13 | 12 | eqeq2d 2632 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥)) |
14 | 13 | dfoprab3 7224 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} |
15 | 8, 9, 14 | 3eqtrri 2649 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 {copab 4712 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 Fn wfn 5883 –onto→wfo 5886 ‘cfv 5888 {coprab 6651 1st c1st 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-oprab 6654 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: df1stres 29481 |
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