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Mirrors > Home > MPE Home > Th. List > fthres2b | Structured version Visualization version GIF version |
Description: Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fthres2b.a | ⊢ 𝐴 = (Base‘𝐶) |
fthres2b.h | ⊢ 𝐻 = (Hom ‘𝐶) |
fthres2b.r | ⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷)) |
fthres2b.s | ⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆)) |
fthres2b.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
fthres2b.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦))) |
Ref | Expression |
---|---|
fthres2b | ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith (𝐷 ↾cat 𝑅))𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthres2b.a | . . . 4 ⊢ 𝐴 = (Base‘𝐶) | |
2 | fthres2b.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | fthres2b.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷)) | |
4 | fthres2b.s | . . . 4 ⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆)) | |
5 | fthres2b.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | fthres2b.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦))) | |
7 | 1, 2, 3, 4, 5, 6 | funcres2b 17015 | . . 3 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺)) |
8 | 7 | anbi1d 620 | . 2 ⊢ (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦)) ↔ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦)))) |
9 | 1 | isfth 17032 | . 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
10 | 1 | isfth 17032 | . 2 ⊢ (𝐹(𝐶 Faith (𝐷 ↾cat 𝑅))𝐺 ↔ (𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
11 | 8, 9, 10 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith (𝐷 ↾cat 𝑅))𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3082 class class class wbr 4923 × cxp 5398 ◡ccnv 5399 Fun wfun 6176 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 Hom chom 16422 ↾cat cresc 16926 Subcatcsubc 16927 Func cfunc 16972 Faith cfth 17021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-hom 16435 df-cco 16436 df-cat 16787 df-cid 16788 df-homf 16789 df-ssc 16928 df-resc 16929 df-subc 16930 df-func 16976 df-fth 17023 |
This theorem is referenced by: (None) |
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