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Mirrors > Home > MPE Home > Th. List > fthres2c | 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, 30-Jan-2017.) |
Ref | Expression |
---|---|
fthres2c.a | ⊢ 𝐴 = (Base‘𝐶) |
fthres2c.e | ⊢ 𝐸 = (𝐷 ↾s 𝑆) |
fthres2c.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
fthres2c.r | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
fthres2c.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
Ref | Expression |
---|---|
fthres2c | ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith 𝐸)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthres2c.a | . . . 4 ⊢ 𝐴 = (Base‘𝐶) | |
2 | fthres2c.e | . . . 4 ⊢ 𝐸 = (𝐷 ↾s 𝑆) | |
3 | fthres2c.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
4 | fthres2c.r | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | fthres2c.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | 1, 2, 3, 4, 5 | funcres2c 17041 | . . 3 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺)) |
7 | 6 | anbi1d 621 | . 2 ⊢ (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦)) ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦)))) |
8 | 1 | isfth 17054 | . 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
9 | 1 | isfth 17054 | . 2 ⊢ (𝐹(𝐶 Faith 𝐸)𝐺 ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
10 | 7, 8, 9 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith 𝐸)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 class class class wbr 4925 ◡ccnv 5402 Fun wfun 6179 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 ↾s cress 16338 Catccat 16805 Func cfunc 16994 Faith cfth 17043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-hom 16443 df-cco 16444 df-cat 16809 df-cid 16810 df-homf 16811 df-comf 16812 df-ssc 16950 df-resc 16951 df-subc 16952 df-func 16998 df-fth 17045 |
This theorem is referenced by: ffthres2c 17080 |
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