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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 | ā¢ {āØāØš„, š¦ā©, š§ā© ā£ š§ = š„} = (1st ā¾ (V Ć V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 7942 | . . . . . 6 ā¢ 1st :VāontoāV | |
2 | fofn 6759 | . . . . . 6 ā¢ (1st :VāontoāV ā 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ā¢ 1st Fn V |
4 | dffn5 6902 | . . . . 5 ā¢ (1st Fn V ā 1st = (š¤ ā V ā¦ (1st āš¤))) | |
5 | 3, 4 | mpbi 229 | . . . 4 ā¢ 1st = (š¤ ā V ā¦ (1st āš¤)) |
6 | mptv 5222 | . . . 4 ā¢ (š¤ ā V ā¦ (1st āš¤)) = {āØš¤, š§ā© ā£ š§ = (1st āš¤)} | |
7 | 5, 6 | eqtri 2761 | . . 3 ā¢ 1st = {āØš¤, š§ā© ā£ š§ = (1st āš¤)} |
8 | 7 | reseq1i 5934 | . 2 ā¢ (1st ā¾ (V Ć V)) = ({āØš¤, š§ā© ā£ š§ = (1st āš¤)} ā¾ (V Ć V)) |
9 | resopab 5989 | . 2 ā¢ ({āØš¤, š§ā© ā£ š§ = (1st āš¤)} ā¾ (V Ć V)) = {āØš¤, š§ā© ā£ (š¤ ā (V Ć V) ā§ š§ = (1st āš¤))} | |
10 | vex 3448 | . . . . 5 ā¢ š„ ā V | |
11 | vex 3448 | . . . . 5 ā¢ š¦ ā V | |
12 | 10, 11 | op1std 7932 | . . . 4 ā¢ (š¤ = āØš„, š¦ā© ā (1st āš¤) = š„) |
13 | 12 | eqeq2d 2744 | . . 3 ā¢ (š¤ = āØš„, š¦ā© ā (š§ = (1st āš¤) ā š§ = š„)) |
14 | 13 | dfoprab3 7987 | . 2 ā¢ {āØš¤, š§ā© ā£ (š¤ ā (V Ć V) ā§ š§ = (1st āš¤))} = {āØāØš„, š¦ā©, š§ā© ā£ š§ = š„} |
15 | 8, 9, 14 | 3eqtrri 2766 | 1 ā¢ {āØāØš„, š¦ā©, š§ā© ā£ š§ = š„} = (1st ā¾ (V Ć V)) |
Colors of variables: wff setvar class |
Syntax hints: ā§ wa 397 = wceq 1542 ā wcel 2107 Vcvv 3444 āØcop 4593 {copab 5168 ā¦ cmpt 5189 Ć cxp 5632 ā¾ cres 5636 Fn wfn 6492 āontoāwfo 6495 ācfv 6497 {coprab 7359 1st c1st 7920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-oprab 7362 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: df1stres 31664 |
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