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Mirrors > Home > MPE Home > Th. List > coapm | Structured version Visualization version Unicode version |
Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
coapm.o | compa |
coapm.a | Nat |
Ref | Expression |
---|---|
coapm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coapm.o | . . . . . 6 compa | |
2 | coapm.a | . . . . . 6 Nat | |
3 | eqid 2622 | . . . . . 6 comp comp | |
4 | 1, 2, 3 | coafval 16714 | . . . . 5 coda coda compcoda |
5 | 4 | mpt2fun 6762 | . . . 4 |
6 | funfn 5918 | . . . 4 | |
7 | 5, 6 | mpbi 220 | . . 3 |
8 | 1, 2 | dmcoass 16716 | . . . . . . . . 9 |
9 | 8 | sseli 3599 | . . . . . . . 8 |
10 | 1st2nd2 7205 | . . . . . . . 8 | |
11 | 9, 10 | syl 17 | . . . . . . 7 |
12 | 11 | fveq2d 6195 | . . . . . 6 |
13 | df-ov 6653 | . . . . . 6 | |
14 | 12, 13 | syl6eqr 2674 | . . . . 5 |
15 | eqid 2622 | . . . . . . 7 Homa Homa | |
16 | 2, 15 | homarw 16696 | . . . . . 6 Homacoda |
17 | id 22 | . . . . . . . . . . . . 13 | |
18 | 11, 17 | eqeltrrd 2702 | . . . . . . . . . . . 12 |
19 | df-br 4654 | . . . . . . . . . . . 12 | |
20 | 18, 19 | sylibr 224 | . . . . . . . . . . 11 |
21 | 1, 2 | eldmcoa 16715 | . . . . . . . . . . 11 coda |
22 | 20, 21 | sylib 208 | . . . . . . . . . 10 coda |
23 | 22 | simp1d 1073 | . . . . . . . . 9 |
24 | 2, 15 | arwhoma 16695 | . . . . . . . . 9 Homacoda |
25 | 23, 24 | syl 17 | . . . . . . . 8 Homacoda |
26 | 22 | simp3d 1075 | . . . . . . . . 9 coda |
27 | 26 | oveq2d 6666 | . . . . . . . 8 Homacoda Homa |
28 | 25, 27 | eleqtrd 2703 | . . . . . . 7 Homa |
29 | 22 | simp2d 1074 | . . . . . . . 8 |
30 | 2, 15 | arwhoma 16695 | . . . . . . . 8 Homacoda |
31 | 29, 30 | syl 17 | . . . . . . 7 Homacoda |
32 | 1, 15, 28, 31 | coahom 16720 | . . . . . 6 Homacoda |
33 | 16, 32 | sseldi 3601 | . . . . 5 |
34 | 14, 33 | eqeltrd 2701 | . . . 4 |
35 | 34 | rgen 2922 | . . 3 |
36 | ffnfv 6388 | . . 3 | |
37 | 7, 35, 36 | mpbir2an 955 | . 2 |
38 | fvex 6201 | . . . 4 Nat | |
39 | 2, 38 | eqeltri 2697 | . . 3 |
40 | 39, 39 | xpex 6962 | . . 3 |
41 | 39, 40 | elpm2 7889 | . 2 |
42 | 37, 8, 41 | mpbir2an 955 | 1 |
Colors of variables: wff setvar class |
Syntax hints: w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 cvv 3200 wss 3574 cop 4183 cotp 4185 class class class wbr 4653 cxp 5112 cdm 5114 wfun 5882 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 cpm 7858 compcco 15953 cdoma 16670 codaccoda 16671 Natcarw 16672 Homachoma 16673 compaccoa 16704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-pm 7860 df-cat 16329 df-doma 16674 df-coda 16675 df-homa 16676 df-arw 16677 df-coa 16706 |
This theorem is referenced by: (None) |
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