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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 7991 | . . . . . 6 ⢠1st :VāontoāV | |
2 | fofn 6800 | . . . . . 6 ⢠(1st :VāontoāV ā 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⢠1st Fn V |
4 | dffn5 6943 | . . . . 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 5257 | . . . 4 ⢠(š¤ ā V ⦠(1st āš¤)) = {āØš¤, š§ā© ⣠š§ = (1st āš¤)} | |
7 | 5, 6 | eqtri 2754 | . . 3 ⢠1st = {āØš¤, š§ā© ⣠š§ = (1st āš¤)} |
8 | 7 | reseq1i 5970 | . 2 ⢠(1st ā¾ (V Ć V)) = ({āØš¤, š§ā© ⣠š§ = (1st āš¤)} ā¾ (V Ć V)) |
9 | resopab 6027 | . 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 3472 | . . . . 5 ⢠š„ ā V | |
11 | vex 3472 | . . . . 5 ⢠š¦ ā V | |
12 | 10, 11 | op1std 7981 | . . . 4 ⢠(š¤ = āØš„, š¦ā© ā (1st āš¤) = š„) |
13 | 12 | eqeq2d 2737 | . . 3 ⢠(š¤ = āØš„, š¦ā© ā (š§ = (1st āš¤) ā š§ = š„)) |
14 | 13 | dfoprab3 8036 | . 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 2759 | 1 ⢠{āØāØš„, š¦ā©, š§ā© ⣠š§ = š„} = (1st ā¾ (V Ć V)) |
Colors of variables: wff setvar class |
Syntax hints: ā§ wa 395 = wceq 1533 ā wcel 2098 Vcvv 3468 āØcop 4629 {copab 5203 ⦠cmpt 5224 Ć cxp 5667 ā¾ cres 5671 Fn wfn 6531 āontoāwfo 6534 ācfv 6536 {coprab 7405 1st c1st 7969 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-oprab 7408 df-1st 7971 df-2nd 7972 |
This theorem is referenced by: df1stres 32430 |
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