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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 7989 | . . . . . 6 β’ 1st :VβontoβV | |
2 | fofn 6798 | . . . . . 6 β’ (1st :VβontoβV β 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ 1st Fn V |
4 | fof 6796 | . . . . . 6 β’ (1st :VβontoβV β 1st :VβΆV) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 β’ 1st :VβΆV |
6 | fnfco 6747 | . . . . 5 β’ ((1st Fn V β§ 1st :VβΆV) β (1st β 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 689 | . . . 4 β’ (1st β 1st ) Fn V |
8 | df-doma 17978 | . . . . 5 β’ doma = (1st β 1st ) | |
9 | 8 | fneq1i 6637 | . . . 4 β’ (doma Fn V β (1st β 1st ) Fn V) |
10 | 7, 9 | mpbir 230 | . . 3 β’ doma Fn V |
11 | ssv 3999 | . . 3 β’ π΄ β V | |
12 | fnssres 6664 | . . 3 β’ ((doma Fn V β§ π΄ β V) β (doma βΎ π΄) Fn π΄) | |
13 | 10, 11, 12 | mp2an 689 | . 2 β’ (doma βΎ π΄) Fn π΄ |
14 | fvres 6901 | . . . 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 18001 | . . . 4 β’ (π₯ β π΄ β (domaβπ₯) β π΅) |
18 | 14, 17 | eqeltrd 2825 | . . 3 β’ (π₯ β π΄ β ((doma βΎ π΄)βπ₯) β π΅) |
19 | 18 | rgen 3055 | . 2 β’ βπ₯ β π΄ ((doma βΎ π΄)βπ₯) β π΅ |
20 | ffnfv 7111 | . 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 708 | 1 β’ (doma βΎ π΄):π΄βΆπ΅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 βwral 3053 Vcvv 3466 β wss 3941 βΎ cres 5669 β ccom 5671 Fn wfn 6529 βΆwf 6530 βontoβwfo 6532 βcfv 6534 1st c1st 7967 Basecbs 17145 domacdoma 17974 Arrowcarw 17976 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-1st 7969 df-2nd 7970 df-doma 17978 df-coda 17979 df-homa 17980 df-arw 17981 |
This theorem is referenced by: (None) |
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