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Mirrors > Home > MPE Home > Th. List > dmaf | Structured version Visualization version GIF version |
Description: The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | β’ π΄ = (ArrowβπΆ) |
arwdm.b | β’ π΅ = (BaseβπΆ) |
Ref | Expression |
---|---|
dmaf | β’ (doma βΎ π΄):π΄βΆπ΅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 7942 | . . . . . 6 β’ 1st :VβontoβV | |
2 | fofn 6759 | . . . . . 6 β’ (1st :VβontoβV β 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ 1st Fn V |
4 | fof 6757 | . . . . . 6 β’ (1st :VβontoβV β 1st :VβΆV) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 β’ 1st :VβΆV |
6 | fnfco 6708 | . . . . 5 β’ ((1st Fn V β§ 1st :VβΆV) β (1st β 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 691 | . . . 4 β’ (1st β 1st ) Fn V |
8 | df-doma 17915 | . . . . 5 β’ doma = (1st β 1st ) | |
9 | 8 | fneq1i 6600 | . . . 4 β’ (doma Fn V β (1st β 1st ) Fn V) |
10 | 7, 9 | mpbir 230 | . . 3 β’ doma Fn V |
11 | ssv 3969 | . . 3 β’ π΄ β V | |
12 | fnssres 6625 | . . 3 β’ ((doma Fn V β§ π΄ β V) β (doma βΎ π΄) Fn π΄) | |
13 | 10, 11, 12 | mp2an 691 | . 2 β’ (doma βΎ π΄) Fn π΄ |
14 | fvres 6862 | . . . 4 β’ (π₯ β π΄ β ((doma βΎ π΄)βπ₯) = (domaβπ₯)) | |
15 | arwrcl.a | . . . . 5 β’ π΄ = (ArrowβπΆ) | |
16 | arwdm.b | . . . . 5 β’ π΅ = (BaseβπΆ) | |
17 | 15, 16 | arwdm 17938 | . . . 4 β’ (π₯ β π΄ β (domaβπ₯) β π΅) |
18 | 14, 17 | eqeltrd 2834 | . . 3 β’ (π₯ β π΄ β ((doma βΎ π΄)βπ₯) β π΅) |
19 | 18 | rgen 3063 | . 2 β’ βπ₯ β π΄ ((doma βΎ π΄)βπ₯) β π΅ |
20 | ffnfv 7067 | . 2 β’ ((doma βΎ π΄):π΄βΆπ΅ β ((doma βΎ π΄) Fn π΄ β§ βπ₯ β π΄ ((doma βΎ π΄)βπ₯) β π΅)) | |
21 | 13, 19, 20 | mpbir2an 710 | 1 β’ (doma βΎ π΄):π΄βΆπ΅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3444 β wss 3911 βΎ cres 5636 β ccom 5638 Fn wfn 6492 βΆwf 6493 βontoβwfo 6495 βcfv 6497 1st c1st 7920 Basecbs 17088 domacdoma 17911 Arrowcarw 17913 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 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-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-1st 7922 df-2nd 7923 df-doma 17915 df-coda 17916 df-homa 17917 df-arw 17918 |
This theorem is referenced by: (None) |
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