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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 17154 | . . 3 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺)) |
7 | 6 | anbi1d 631 | . 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 17167 | . 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
9 | 1 | isfth 17167 | . 2 ⊢ (𝐹(𝐶 Faith 𝐸)𝐺 ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
10 | 7, 8, 9 | 3bitr4g 316 | 1 ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith 𝐸)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 class class class wbr 5052 ◡ccnv 5540 Fun wfun 6335 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 ↾s cress 16467 Catccat 16918 Func cfunc 17107 Faith cfth 17156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-hom 16572 df-cco 16573 df-cat 16922 df-cid 16923 df-homf 16924 df-comf 16925 df-ssc 17063 df-resc 17064 df-subc 17065 df-func 17111 df-fth 17158 |
This theorem is referenced by: ffthres2c 17193 |
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