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Mirrors > Home > MPE Home > Th. List > ffthiso | Structured version Visualization version Unicode version |
Description: A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fthmon.b | |
fthmon.h | |
fthmon.f | Faith |
fthmon.x | |
fthmon.y | |
fthmon.r | |
ffthiso.f | Full |
ffthiso.s | |
ffthiso.t |
Ref | Expression |
---|---|
ffthiso |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthmon.b | . . 3 | |
2 | ffthiso.s | . . 3 | |
3 | ffthiso.t | . . 3 | |
4 | fthmon.f | . . . . 5 Faith | |
5 | fthfunc 16567 | . . . . . 6 Faith | |
6 | 5 | ssbri 4697 | . . . . 5 Faith |
7 | 4, 6 | syl 17 | . . . 4 |
8 | 7 | adantr 481 | . . 3 |
9 | fthmon.x | . . . 4 | |
10 | 9 | adantr 481 | . . 3 |
11 | fthmon.y | . . . 4 | |
12 | 11 | adantr 481 | . . 3 |
13 | simpr 477 | . . 3 | |
14 | 1, 2, 3, 8, 10, 12, 13 | funciso 16534 | . 2 |
15 | eqid 2622 | . . . 4 Inv Inv | |
16 | df-br 4654 | . . . . . . . 8 | |
17 | 7, 16 | sylib 208 | . . . . . . 7 |
18 | funcrcl 16523 | . . . . . . 7 | |
19 | 17, 18 | syl 17 | . . . . . 6 |
20 | 19 | simpld 475 | . . . . 5 |
21 | 20 | ad3antrrr 766 | . . . 4 Inv |
22 | 9 | ad3antrrr 766 | . . . 4 Inv |
23 | 11 | ad3antrrr 766 | . . . 4 Inv |
24 | eqid 2622 | . . . . . . . . . . 11 | |
25 | eqid 2622 | . . . . . . . . . . 11 Inv Inv | |
26 | 19 | simprd 479 | . . . . . . . . . . 11 |
27 | 1, 24, 7 | funcf1 16526 | . . . . . . . . . . . 12 |
28 | 27, 9 | ffvelrnd 6360 | . . . . . . . . . . 11 |
29 | 27, 11 | ffvelrnd 6360 | . . . . . . . . . . 11 |
30 | 24, 25, 26, 28, 29, 3 | isoval 16425 | . . . . . . . . . 10 Inv |
31 | 30 | eleq2d 2687 | . . . . . . . . 9 Inv |
32 | 31 | biimpa 501 | . . . . . . . 8 Inv |
33 | 24, 25, 26, 28, 29 | invfun 16424 | . . . . . . . . . 10 Inv |
34 | 33 | adantr 481 | . . . . . . . . 9 Inv |
35 | funfvbrb 6330 | . . . . . . . . 9 Inv Inv InvInv | |
36 | 34, 35 | syl 17 | . . . . . . . 8 Inv InvInv |
37 | 32, 36 | mpbid 222 | . . . . . . 7 InvInv |
38 | 37 | ad2antrr 762 | . . . . . 6 Inv InvInv |
39 | simpr 477 | . . . . . 6 Inv Inv | |
40 | 38, 39 | breqtrd 4679 | . . . . 5 Inv Inv |
41 | fthmon.h | . . . . . 6 | |
42 | 4 | ad3antrrr 766 | . . . . . 6 Inv Faith |
43 | fthmon.r | . . . . . . 7 | |
44 | 43 | ad3antrrr 766 | . . . . . 6 Inv |
45 | simplr 792 | . . . . . 6 Inv | |
46 | 1, 41, 42, 22, 23, 44, 45, 15, 25 | fthinv 16586 | . . . . 5 Inv Inv Inv |
47 | 40, 46 | mpbird 247 | . . . 4 Inv Inv |
48 | 1, 15, 21, 22, 23, 2, 47 | inviso1 16426 | . . 3 Inv |
49 | eqid 2622 | . . . 4 | |
50 | ffthiso.f | . . . . 5 Full | |
51 | 50 | adantr 481 | . . . 4 Full |
52 | 11 | adantr 481 | . . . 4 |
53 | 9 | adantr 481 | . . . 4 |
54 | 24, 49, 3, 26, 29, 28 | isohom 16436 | . . . . . 6 |
55 | 54 | adantr 481 | . . . . 5 |
56 | 24, 25, 26, 28, 29, 3 | invf 16428 | . . . . . 6 Inv |
57 | 56 | ffvelrnda 6359 | . . . . 5 Inv |
58 | 55, 57 | sseldd 3604 | . . . 4 Inv |
59 | 1, 49, 41, 51, 52, 53, 58 | fulli 16573 | . . 3 Inv |
60 | 48, 59 | r19.29a 3078 | . 2 |
61 | 14, 60 | impbida 877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wss 3574 cop 4183 class class class wbr 4653 cdm 5114 wfun 5882 cfv 5888 (class class class)co 6650 cbs 15857 chom 15952 ccat 16325 Invcinv 16405 ciso 16406 cfunc 16514 Full cful 16562 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-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-rmo 2920 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-sect 16407 df-inv 16408 df-iso 16409 df-func 16518 df-full 16564 df-fth 16565 |
This theorem is referenced by: (None) |
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