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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 8007 | . . . . . 6 β’ 1st :VβontoβV | |
2 | fofn 6807 | . . . . . 6 β’ (1st :VβontoβV β 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ 1st Fn V |
4 | fof 6805 | . . . . . 6 β’ (1st :VβontoβV β 1st :VβΆV) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 β’ 1st :VβΆV |
6 | fnfco 6756 | . . . . 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 18006 | . . . . 5 β’ doma = (1st β 1st ) | |
9 | 8 | fneq1i 6645 | . . . 4 β’ (doma Fn V β (1st β 1st ) Fn V) |
10 | 7, 9 | mpbir 230 | . . 3 β’ doma Fn V |
11 | ssv 4002 | . . 3 β’ π΄ β V | |
12 | fnssres 6672 | . . 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 6910 | . . . 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 18029 | . . . 4 β’ (π₯ β π΄ β (domaβπ₯) β π΅) |
18 | 14, 17 | eqeltrd 2829 | . . 3 β’ (π₯ β π΄ β ((doma βΎ π΄)βπ₯) β π΅) |
19 | 18 | rgen 3059 | . 2 β’ βπ₯ β π΄ ((doma βΎ π΄)βπ₯) β π΅ |
20 | ffnfv 7123 | . 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 1534 β wcel 2099 βwral 3057 Vcvv 3470 β wss 3945 βΎ cres 5674 β ccom 5676 Fn wfn 6537 βΆwf 6538 βontoβwfo 6540 βcfv 6542 1st c1st 7985 Basecbs 17173 domacdoma 18002 Arrowcarw 18004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-1st 7987 df-2nd 7988 df-doma 18006 df-coda 18007 df-homa 18008 df-arw 18009 |
This theorem is referenced by: (None) |
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