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Mirrors > Home > MPE Home > Th. List > fthpropd | Structured version Visualization version Unicode version |
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fullpropd.1 | f f |
fullpropd.2 | compf compf |
fullpropd.3 | f f |
fullpropd.4 | compf compf |
fullpropd.a | |
fullpropd.b | |
fullpropd.c | |
fullpropd.d |
Ref | Expression |
---|---|
fthpropd | Faith Faith |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfth 16569 | . 2 Faith | |
2 | relfth 16569 | . 2 Faith | |
3 | fullpropd.1 | . . . . . 6 f f | |
4 | fullpropd.2 | . . . . . 6 compf compf | |
5 | fullpropd.3 | . . . . . 6 f f | |
6 | fullpropd.4 | . . . . . 6 compf compf | |
7 | fullpropd.a | . . . . . 6 | |
8 | fullpropd.b | . . . . . 6 | |
9 | fullpropd.c | . . . . . 6 | |
10 | fullpropd.d | . . . . . 6 | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | funcpropd 16560 | . . . . 5 |
12 | 11 | breqd 4664 | . . . 4 |
13 | 3 | homfeqbas 16356 | . . . . 5 |
14 | 13 | raleqdv 3144 | . . . . 5 |
15 | 13, 14 | raleqbidv 3152 | . . . 4 |
16 | 12, 15 | anbi12d 747 | . . 3 |
17 | eqid 2622 | . . . 4 | |
18 | 17 | isfth 16574 | . . 3 Faith |
19 | eqid 2622 | . . . 4 | |
20 | 19 | isfth 16574 | . . 3 Faith |
21 | 16, 18, 20 | 3bitr4g 303 | . 2 Faith Faith |
22 | 1, 2, 21 | eqbrrdiv 5218 | 1 Faith Faith |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 class class class wbr 4653 ccnv 5113 wfun 5882 cfv 5888 (class class class)co 6650 cbs 15857 f chomf 16327 compfccomf 16328 cfunc 16514 Faith cfth 16563 |
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-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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-ixp 7909 df-cat 16329 df-cid 16330 df-homf 16331 df-comf 16332 df-func 16518 df-fth 16565 |
This theorem is referenced by: (None) |
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