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Mirrors > Home > MPE Home > Th. List > xpcco2 | Structured version Visualization version Unicode version |
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
xpcco2.t | c |
xpcco2.x | |
xpcco2.y | |
xpcco2.h | |
xpcco2.j | |
xpcco2.m | |
xpcco2.n | |
xpcco2.p | |
xpcco2.q | |
xpcco2.o1 | comp |
xpcco2.o2 | comp |
xpcco2.o | comp |
xpcco2.r | |
xpcco2.s | |
xpcco2.f | |
xpcco2.g | |
xpcco2.k | |
xpcco2.l |
Ref | Expression |
---|---|
xpcco2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpcco2.t | . . 3 c | |
2 | xpcco2.x | . . . 4 | |
3 | xpcco2.y | . . . 4 | |
4 | 1, 2, 3 | xpcbas 16818 | . . 3 |
5 | eqid 2622 | . . 3 | |
6 | xpcco2.o1 | . . 3 comp | |
7 | xpcco2.o2 | . . 3 comp | |
8 | xpcco2.o | . . 3 comp | |
9 | xpcco2.m | . . . 4 | |
10 | xpcco2.n | . . . 4 | |
11 | opelxpi 5148 | . . . 4 | |
12 | 9, 10, 11 | syl2anc 693 | . . 3 |
13 | xpcco2.p | . . . 4 | |
14 | xpcco2.q | . . . 4 | |
15 | opelxpi 5148 | . . . 4 | |
16 | 13, 14, 15 | syl2anc 693 | . . 3 |
17 | xpcco2.r | . . . 4 | |
18 | xpcco2.s | . . . 4 | |
19 | opelxpi 5148 | . . . 4 | |
20 | 17, 18, 19 | syl2anc 693 | . . 3 |
21 | xpcco2.f | . . . . 5 | |
22 | xpcco2.g | . . . . 5 | |
23 | opelxpi 5148 | . . . . 5 | |
24 | 21, 22, 23 | syl2anc 693 | . . . 4 |
25 | xpcco2.h | . . . . 5 | |
26 | xpcco2.j | . . . . 5 | |
27 | 1, 2, 3, 25, 26, 9, 10, 13, 14, 5 | xpchom2 16826 | . . . 4 |
28 | 24, 27 | eleqtrrd 2704 | . . 3 |
29 | xpcco2.k | . . . . 5 | |
30 | xpcco2.l | . . . . 5 | |
31 | opelxpi 5148 | . . . . 5 | |
32 | 29, 30, 31 | syl2anc 693 | . . . 4 |
33 | 1, 2, 3, 25, 26, 13, 14, 17, 18, 5 | xpchom2 16826 | . . . 4 |
34 | 32, 33 | eleqtrrd 2704 | . . 3 |
35 | 1, 4, 5, 6, 7, 8, 12, 16, 20, 28, 34 | xpcco 16823 | . 2 |
36 | op1stg 7180 | . . . . . . 7 | |
37 | 9, 10, 36 | syl2anc 693 | . . . . . 6 |
38 | op1stg 7180 | . . . . . . 7 | |
39 | 13, 14, 38 | syl2anc 693 | . . . . . 6 |
40 | 37, 39 | opeq12d 4410 | . . . . 5 |
41 | op1stg 7180 | . . . . . 6 | |
42 | 17, 18, 41 | syl2anc 693 | . . . . 5 |
43 | 40, 42 | oveq12d 6668 | . . . 4 |
44 | op1stg 7180 | . . . . 5 | |
45 | 29, 30, 44 | syl2anc 693 | . . . 4 |
46 | op1stg 7180 | . . . . 5 | |
47 | 21, 22, 46 | syl2anc 693 | . . . 4 |
48 | 43, 45, 47 | oveq123d 6671 | . . 3 |
49 | op2ndg 7181 | . . . . . . 7 | |
50 | 9, 10, 49 | syl2anc 693 | . . . . . 6 |
51 | op2ndg 7181 | . . . . . . 7 | |
52 | 13, 14, 51 | syl2anc 693 | . . . . . 6 |
53 | 50, 52 | opeq12d 4410 | . . . . 5 |
54 | op2ndg 7181 | . . . . . 6 | |
55 | 17, 18, 54 | syl2anc 693 | . . . . 5 |
56 | 53, 55 | oveq12d 6668 | . . . 4 |
57 | op2ndg 7181 | . . . . 5 | |
58 | 29, 30, 57 | syl2anc 693 | . . . 4 |
59 | op2ndg 7181 | . . . . 5 | |
60 | 21, 22, 59 | syl2anc 693 | . . . 4 |
61 | 56, 58, 60 | oveq123d 6671 | . . 3 |
62 | 48, 61 | opeq12d 4410 | . 2 |
63 | 35, 62 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cop 4183 cxp 5112 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 cbs 15857 chom 15952 compcco 15953 c cxpc 16808 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-nel 2898 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-xpc 16812 |
This theorem is referenced by: prfcl 16843 evlfcllem 16861 curf1cl 16868 curf2cl 16871 curfcl 16872 uncfcurf 16879 hofcl 16899 |
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