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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 17929 | . . . . 5 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
5 | 4 | ssbri 5198 | . . . 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 17894 | . 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) |
8 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | eqid 2726 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
10 | eqid 2726 | . . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
11 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Faith 𝐷)𝐺) |
12 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
13 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
14 | 8, 9, 10, 11, 12, 13 | fthf1 17939 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)–1-1→((𝐹‘𝑦)(Hom ‘𝐷)(𝐹‘𝑥))) |
15 | df-f1 6559 | . . . . . 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 495 | . . . . 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 8256 | . . . . . 6 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥) | |
19 | 18 | cnveqi 5881 | . . . . 5 ⊢ ◡(𝑥tpos 𝐺𝑦) = ◡(𝑦𝐺𝑥) |
20 | 19 | funeqi 6580 | . . . 4 ⊢ (Fun ◡(𝑥tpos 𝐺𝑦) ↔ Fun ◡(𝑦𝐺𝑥)) |
21 | 17, 20 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → Fun ◡(𝑥tpos 𝐺𝑦)) |
22 | 21 | ralrimivva 3191 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)Fun ◡(𝑥tpos 𝐺𝑦)) |
23 | 1, 8 | oppcbas 17732 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
24 | 23 | isfth 17936 | . 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 581 | 1 ⊢ (𝜑 → 𝐹(𝑂 Faith 𝑃)tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 class class class wbr 5153 ◡ccnv 5681 Fun wfun 6548 ⟶wf 6550 –1-1→wf1 6551 ‘cfv 6554 (class class class)co 7424 tpos ctpos 8240 Basecbs 17213 Hom chom 17277 oppCatcoppc 17724 Func cfunc 17873 Faith cfth 17925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-hom 17290 df-cco 17291 df-cat 17681 df-cid 17682 df-oppc 17725 df-func 17877 df-fth 17927 |
This theorem is referenced by: ffthoppc 17946 fthepi 17950 |
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