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Mirrors > Home > MPE Home > Th. List > oyoncl | Structured version Visualization version Unicode version |
Description: The opposite Yoneda
embedding is a functor from oppCat![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
oyoncl.o |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.y |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.s |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.u |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.h |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
oyoncl.q |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
oyoncl |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oyoncl.y |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | oyoncl.c |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | oyoncl.o |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 3 | oppccat 15627 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 2, 4 | syl 17 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | eqid 2451 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | oyoncl.s |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | eqid 2451 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | oyoncl.u |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | eqid 2451 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 3, 10 | oppchomf 15625 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | rneqi 5061 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | relxp 4942 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | eqid 2451 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 10, 14 | homffn 15598 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | fndm 5675 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | ax-mp 5 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | releqi 4918 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 13, 18 | mpbir 213 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | rntpos 6986 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | ax-mp 5 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 12, 21 | eqtr3i 2475 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | oyoncl.h |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl5eqss 3476 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 1, 5, 6, 7, 8, 9, 24 | yoncl 16147 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | oyoncl.q |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 3 | 2oppchomf 15629 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 3 | 2oppccomf 15630 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | a1i 11 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | eqidd 2452 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | eqidd 2452 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 6 | oppccat 15627 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 5, 33 | syl 17 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 7 | setccat 15980 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 9, 35 | syl 17 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 28, 30, 31, 32, 2, 34, 36, 36 | fucpropd 15882 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 26, 37 | syl5eq 2497 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 38 | oveq2d 6306 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 25, 39 | eleqtrrd 2532 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-fal 1450 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-tpos 6973 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-map 7474 df-ixp 7523 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-7 10673 df-8 10674 df-9 10675 df-10 10676 df-n0 10870 df-z 10938 df-dec 11052 df-uz 11160 df-fz 11785 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-hom 15214 df-cco 15215 df-cat 15574 df-cid 15575 df-homf 15576 df-comf 15577 df-oppc 15617 df-func 15763 df-nat 15848 df-fuc 15849 df-setc 15971 df-xpc 16057 df-curf 16099 df-hof 16135 df-yon 16136 |
This theorem is referenced by: oyon1cl 16156 |
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