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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 | ⢠{āØāØš„, š¦ā©, š§ā© ⣠š§ = š„} = (1st ā¾ (V Ć V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 8017 | . . . . . 6 ⢠1st :VāontoāV | |
2 | fofn 6816 | . . . . . 6 ⢠(1st :VāontoāV ā 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⢠1st Fn V |
4 | dffn5 6960 | . . . . 5 ⢠(1st Fn V ā 1st = (š¤ ā V ⦠(1st āš¤))) | |
5 | 3, 4 | mpbi 229 | . . . 4 ⢠1st = (š¤ ā V ⦠(1st āš¤)) |
6 | mptv 5266 | . . . 4 ⢠(š¤ ā V ⦠(1st āš¤)) = {āØš¤, š§ā© ⣠š§ = (1st āš¤)} | |
7 | 5, 6 | eqtri 2755 | . . 3 ⢠1st = {āØš¤, š§ā© ⣠š§ = (1st āš¤)} |
8 | 7 | reseq1i 5983 | . 2 ⢠(1st ā¾ (V Ć V)) = ({āØš¤, š§ā© ⣠š§ = (1st āš¤)} ā¾ (V Ć V)) |
9 | resopab 6041 | . 2 ⢠({āØš¤, š§ā© ⣠š§ = (1st āš¤)} ā¾ (V Ć V)) = {āØš¤, š§ā© ⣠(š¤ ā (V Ć V) ā§ š§ = (1st āš¤))} | |
10 | vex 3475 | . . . . 5 ⢠š„ ā V | |
11 | vex 3475 | . . . . 5 ⢠š¦ ā V | |
12 | 10, 11 | op1std 8007 | . . . 4 ⢠(š¤ = āØš„, š¦ā© ā (1st āš¤) = š„) |
13 | 12 | eqeq2d 2738 | . . 3 ⢠(š¤ = āØš„, š¦ā© ā (š§ = (1st āš¤) ā š§ = š„)) |
14 | 13 | dfoprab3 8062 | . 2 ⢠{āØš¤, š§ā© ⣠(š¤ ā (V Ć V) ā§ š§ = (1st āš¤))} = {āØāØš„, š¦ā©, š§ā© ⣠š§ = š„} |
15 | 8, 9, 14 | 3eqtrri 2760 | 1 ⢠{āØāØš„, š¦ā©, š§ā© ⣠š§ = š„} = (1st ā¾ (V Ć V)) |
Colors of variables: wff setvar class |
Syntax hints: ā§ wa 394 = wceq 1533 ā wcel 2098 Vcvv 3471 āØcop 4636 {copab 5212 ⦠cmpt 5233 Ć cxp 5678 ā¾ cres 5682 Fn wfn 6546 āontoāwfo 6549 ācfv 6551 {coprab 7425 1st c1st 7995 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fo 6557 df-fv 6559 df-oprab 7428 df-1st 7997 df-2nd 7998 |
This theorem is referenced by: df1stres 32501 |
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