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Mirrors > Home > MPE Home > Th. List > fthoppc | Structured version Visualization version GIF version |
Description: The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fulloppc.o | ⊢ 𝑂 = (oppCat‘𝐶) |
fulloppc.p | ⊢ 𝑃 = (oppCat‘𝐷) |
fthoppc.f | ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
Ref | Expression |
---|---|
fthoppc | ⊢ (𝜑 → 𝐹(𝑂 Faith 𝑃)tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fulloppc.o | . . 3 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | fulloppc.p | . . 3 ⊢ 𝑃 = (oppCat‘𝐷) | |
3 | fthoppc.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
4 | fthfunc 16567 | . . . . 5 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
5 | 4 | ssbri 4697 | . . . 4 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
7 | 1, 2, 6 | funcoppc 16535 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) |
8 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2622 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
10 | eqid 2622 | . . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
11 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Faith 𝐷)𝐺) |
12 | simprr 796 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
13 | simprl 794 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
14 | 8, 9, 10, 11, 12, 13 | fthf1 16577 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–1-1→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
15 | df-f1 5893 | . . . . . 6 ⊢ ((𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–1-1→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) ↔ ((𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) ∧ Fun ◡(𝑦𝐺𝑥))) | |
16 | 15 | simprbi 480 | . . . . 5 ⊢ ((𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–1-1→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥)) → Fun ◡(𝑦𝐺𝑥)) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → Fun ◡(𝑦𝐺𝑥)) |
18 | ovtpos 7367 | . . . . . 6 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥) | |
19 | 18 | cnveqi 5297 | . . . . 5 ⊢ ◡(𝑥tpos 𝐺𝑦) = ◡(𝑦𝐺𝑥) |
20 | 19 | funeqi 5909 | . . . 4 ⊢ (Fun ◡(𝑥tpos 𝐺𝑦) ↔ Fun ◡(𝑦𝐺𝑥)) |
21 | 17, 20 | sylibr 224 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → Fun ◡(𝑥tpos 𝐺𝑦)) |
22 | 21 | ralrimivva 2971 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)Fun ◡(𝑥tpos 𝐺𝑦)) |
23 | 1, 8 | oppcbas 16378 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
24 | 23 | isfth 16574 | . 2 ⊢ (𝐹(𝑂 Faith 𝑃)tpos 𝐺 ↔ (𝐹(𝑂 Func 𝑃)tpos 𝐺 ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)Fun ◡(𝑥tpos 𝐺𝑦))) |
25 | 7, 22, 24 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝐹(𝑂 Faith 𝑃)tpos 𝐺) |
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 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 tpos ctpos 7351 Basecbs 15857 Hom chom 15952 oppCatcoppc 16371 Func 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 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-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-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-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-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-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-hom 15966 df-cco 15967 df-cat 16329 df-cid 16330 df-oppc 16372 df-func 16518 df-fth 16565 |
This theorem is referenced by: ffthoppc 16584 fthepi 16588 |
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