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Mirrors > Home > MPE Home > Th. List > dmcoass | Structured version Visualization version GIF version |
Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
coafval.o | β’ Β· = (compaβπΆ) |
coafval.a | β’ π΄ = (ArrowβπΆ) |
Ref | Expression |
---|---|
dmcoass | β’ dom Β· β (π΄ Γ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coafval.o | . . . 4 β’ Β· = (compaβπΆ) | |
2 | coafval.a | . . . 4 β’ π΄ = (ArrowβπΆ) | |
3 | eqid 2732 | . . . 4 β’ (compβπΆ) = (compβπΆ) | |
4 | 1, 2, 3 | coafval 18018 | . . 3 β’ Β· = (π β π΄, π β {β β π΄ β£ (codaββ) = (domaβπ)} β¦ β¨(domaβπ), (codaβπ), ((2nd βπ)(β¨(domaβπ), (domaβπ)β©(compβπΆ)(codaβπ))(2nd βπ))β©) |
5 | 4 | dmmpossx 8054 | . 2 β’ dom Β· β βͺ π β π΄ ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) |
6 | iunss 5048 | . . 3 β’ (βͺ π β π΄ ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) β (π΄ Γ π΄) β βπ β π΄ ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) β (π΄ Γ π΄)) | |
7 | snssi 4811 | . . . 4 β’ (π β π΄ β {π} β π΄) | |
8 | ssrab2 4077 | . . . 4 β’ {β β π΄ β£ (codaββ) = (domaβπ)} β π΄ | |
9 | xpss12 5691 | . . . 4 β’ (({π} β π΄ β§ {β β π΄ β£ (codaββ) = (domaβπ)} β π΄) β ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) β (π΄ Γ π΄)) | |
10 | 7, 8, 9 | sylancl 586 | . . 3 β’ (π β π΄ β ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) β (π΄ Γ π΄)) |
11 | 6, 10 | mprgbir 3068 | . 2 β’ βͺ π β π΄ ({π} Γ {β β π΄ β£ (codaββ) = (domaβπ)}) β (π΄ Γ π΄) |
12 | 5, 11 | sstri 3991 | 1 β’ dom Β· β (π΄ Γ π΄) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 {crab 3432 β wss 3948 {csn 4628 β¨cop 4634 β¨cotp 4636 βͺ ciun 4997 Γ cxp 5674 dom cdm 5676 βcfv 6543 (class class class)co 7411 2nd c2nd 7976 compcco 17213 domacdoma 17974 codaccoda 17975 Arrowcarw 17976 compaccoa 18008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-arw 17981 df-coa 18010 |
This theorem is referenced by: coapm 18025 |
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