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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 16384 | . . 3 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺)) |
7 | 6 | anbi1d 737 | . 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 16397 | . 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
9 | 1 | isfth 16397 | . 2 ⊢ (𝐹(𝐶 Faith 𝐸)𝐺 ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 Fun ◡(𝑥𝐺𝑦))) |
10 | 7, 8, 9 | 3bitr4g 302 | 1 ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith 𝐸)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ◡ccnv 5037 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Catccat 16148 Func cfunc 16337 Faith cfth 16386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-hom 15793 df-cco 15794 df-cat 16152 df-cid 16153 df-homf 16154 df-comf 16155 df-ssc 16293 df-resc 16294 df-subc 16295 df-func 16341 df-fth 16388 |
This theorem is referenced by: ffthres2c 16423 |
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