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Mirrors > Home > MPE Home > Th. List > fthpropd | Structured version Visualization version GIF version |
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fullpropd.1 | ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) |
fullpropd.2 | ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) |
fullpropd.3 | ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) |
fullpropd.4 | ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) |
fullpropd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fullpropd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fullpropd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
fullpropd.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
Ref | Expression |
---|---|
fthpropd | ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfth 17031 | . 2 ⊢ Rel (𝐴 Faith 𝐶) | |
2 | relfth 17031 | . 2 ⊢ Rel (𝐵 Faith 𝐷) | |
3 | fullpropd.1 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) | |
4 | fullpropd.2 | . . . . . 6 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) | |
5 | fullpropd.3 | . . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | |
6 | fullpropd.4 | . . . . . 6 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) | |
7 | fullpropd.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | fullpropd.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
9 | fullpropd.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
10 | fullpropd.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | funcpropd 17022 | . . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷)) |
12 | 11 | breqd 4934 | . . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔)) |
13 | 3 | homfeqbas 16818 | . . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵)) |
14 | 13 | raleqdv 3349 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
15 | 13, 14 | raleqbidv 3335 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
16 | 12, 15 | anbi12d 621 | . . 3 ⊢ (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦)) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))) |
17 | eqid 2772 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
18 | 17 | isfth 17036 | . . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦))) |
19 | eqid 2772 | . . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵) | |
20 | 19 | isfth 17036 | . . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
21 | 16, 18, 20 | 3bitr4g 306 | . 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔)) |
22 | 1, 2, 21 | eqbrrdiv 5511 | 1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3082 class class class wbr 4923 ◡ccnv 5400 Fun wfun 6176 ‘cfv 6182 (class class class)co 6970 Basecbs 16333 Homf chomf 16789 compfccomf 16790 Func cfunc 16976 Faith cfth 17025 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 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-1st 7495 df-2nd 7496 df-map 8202 df-ixp 8254 df-cat 16791 df-cid 16792 df-homf 16793 df-comf 16794 df-func 16980 df-fth 17027 |
This theorem is referenced by: (None) |
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