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Mirrors > Home > MPE Home > Th. List > monpropd | Structured version Visualization version Unicode version |
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
monpropd.3 | f f |
monpropd.4 | compf compf |
monpropd.c | |
monpropd.d |
Ref | Expression |
---|---|
monpropd | Mono Mono |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . . . . . . 12 | |
2 | eqid 2622 | . . . . . . . . . . . 12 | |
3 | eqid 2622 | . . . . . . . . . . . 12 | |
4 | monpropd.3 | . . . . . . . . . . . . . 14 f f | |
5 | 4 | ad2antrr 762 | . . . . . . . . . . . . 13 f f |
6 | 5 | ad2antrr 762 | . . . . . . . . . . . 12 f f |
7 | simpr 477 | . . . . . . . . . . . 12 | |
8 | simp-4r 807 | . . . . . . . . . . . 12 | |
9 | 1, 2, 3, 6, 7, 8 | homfeqval 16357 | . . . . . . . . . . 11 |
10 | eqid 2622 | . . . . . . . . . . . 12 comp comp | |
11 | eqid 2622 | . . . . . . . . . . . 12 comp comp | |
12 | 4 | ad5antr 770 | . . . . . . . . . . . 12 f f |
13 | monpropd.4 | . . . . . . . . . . . . 13 compf compf | |
14 | 13 | ad5antr 770 | . . . . . . . . . . . 12 compf compf |
15 | simplr 792 | . . . . . . . . . . . 12 | |
16 | simp-5r 809 | . . . . . . . . . . . 12 | |
17 | simp-4r 807 | . . . . . . . . . . . 12 | |
18 | simpr 477 | . . . . . . . . . . . 12 | |
19 | simpllr 799 | . . . . . . . . . . . 12 | |
20 | 1, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19 | comfeqval 16368 | . . . . . . . . . . 11 comp comp |
21 | 9, 20 | mpteq12dva 4732 | . . . . . . . . . 10 comp comp |
22 | 21 | cnveqd 5298 | . . . . . . . . 9 comp comp |
23 | 22 | funeqd 5910 | . . . . . . . 8 comp comp |
24 | 23 | ralbidva 2985 | . . . . . . 7 comp comp |
25 | 24 | rabbidva 3188 | . . . . . 6 comp comp |
26 | simplr 792 | . . . . . . . 8 | |
27 | simpr 477 | . . . . . . . 8 | |
28 | 1, 2, 3, 5, 26, 27 | homfeqval 16357 | . . . . . . 7 |
29 | 4 | homfeqbas 16356 | . . . . . . . . 9 |
30 | 29 | ad2antrr 762 | . . . . . . . 8 |
31 | 30 | raleqdv 3144 | . . . . . . 7 comp comp |
32 | 28, 31 | rabeqbidv 3195 | . . . . . 6 comp comp |
33 | 25, 32 | eqtrd 2656 | . . . . 5 comp comp |
34 | 33 | 3impa 1259 | . . . 4 comp comp |
35 | 34 | mpt2eq3dva 6719 | . . 3 comp comp |
36 | mpt2eq12 6715 | . . . 4 comp comp | |
37 | 29, 29, 36 | syl2anc 693 | . . 3 comp comp |
38 | 35, 37 | eqtrd 2656 | . 2 comp comp |
39 | eqid 2622 | . . 3 Mono Mono | |
40 | monpropd.c | . . 3 | |
41 | 1, 2, 10, 39, 40 | monfval 16392 | . 2 Mono comp |
42 | eqid 2622 | . . 3 | |
43 | eqid 2622 | . . 3 Mono Mono | |
44 | monpropd.d | . . 3 | |
45 | 42, 3, 11, 43, 44 | monfval 16392 | . 2 Mono comp |
46 | 38, 41, 45 | 3eqtr4d 2666 | 1 Mono Mono |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 crab 2916 cop 4183 cmpt 4729 ccnv 5113 wfun 5882 cfv 5888 (class class class)co 6650 cmpt2 6652 cbs 15857 chom 15952 compcco 15953 ccat 16325 f chomf 16327 compfccomf 16328 Monocmon 16388 |
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-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-homf 16331 df-comf 16332 df-mon 16390 |
This theorem is referenced by: oppcepi 16399 |
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