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Mirrors > Home > MPE Home > Th. List > fullres2c | Structured version Visualization version GIF version |
Description: Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.) |
Ref | Expression |
---|---|
ffthres2c.a | ⊢ 𝐴 = (Base‘𝐶) |
ffthres2c.e | ⊢ 𝐸 = (𝐷 ↾s 𝑆) |
ffthres2c.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
ffthres2c.r | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
ffthres2c.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
Ref | Expression |
---|---|
fullres2c | ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ 𝐹(𝐶 Full 𝐸)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffthres2c.a | . . . 4 ⊢ 𝐴 = (Base‘𝐶) | |
2 | ffthres2c.e | . . . 4 ⊢ 𝐸 = (𝐷 ↾s 𝑆) | |
3 | ffthres2c.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
4 | ffthres2c.r | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | ffthres2c.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | 1, 2, 3, 4, 5 | funcres2c 16874 | . . 3 ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺)) |
7 | eqid 2800 | . . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
8 | 2, 7 | resshom 16392 | . . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸)) |
9 | 4, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸)) |
10 | 9 | oveqd 6896 | . . . . 5 ⊢ (𝜑 → ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦))) |
11 | 10 | eqeq2d 2810 | . . . 4 ⊢ (𝜑 → (ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))) |
12 | 11 | 2ralbidv 3171 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))) |
13 | 6, 12 | anbi12d 625 | . 2 ⊢ (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦))) ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦))))) |
14 | 1, 7 | isfull 16883 | . 2 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))) |
15 | eqid 2800 | . . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
16 | 1, 15 | isfull 16883 | . 2 ⊢ (𝐹(𝐶 Full 𝐸)𝐺 ↔ (𝐹(𝐶 Func 𝐸)𝐺 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)(Hom ‘𝐸)(𝐹‘𝑦)))) |
17 | 13, 14, 16 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ 𝐹(𝐶 Full 𝐸)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3090 class class class wbr 4844 ran crn 5314 ⟶wf 6098 ‘cfv 6102 (class class class)co 6879 Basecbs 16183 ↾s cress 16184 Hom chom 16277 Catccat 16638 Func cfunc 16827 Full cful 16875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-er 7983 df-map 8098 df-pm 8099 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-hom 16290 df-cco 16291 df-cat 16642 df-cid 16643 df-homf 16644 df-comf 16645 df-ssc 16783 df-resc 16784 df-subc 16785 df-func 16831 df-full 16877 |
This theorem is referenced by: ffthres2c 16913 |
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