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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 17370 | . 2 ⊢ Rel (𝐴 Faith 𝐶) | |
2 | relfth 17370 | . 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 17361 | . . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷)) |
12 | 11 | breqd 5050 | . . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔)) |
13 | 3 | homfeqbas 17153 | . . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵)) |
14 | 13 | raleqdv 3315 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
15 | 13, 14 | raleqbidv 3303 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
16 | 12, 15 | anbi12d 634 | . . 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 2736 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
18 | 17 | isfth 17375 | . . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦))) |
19 | eqid 2736 | . . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵) | |
20 | 19 | isfth 17375 | . . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
21 | 16, 18, 20 | 3bitr4g 317 | . 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔)) |
22 | 1, 2, 21 | eqbrrdiv 5649 | 1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 class class class wbr 5039 ◡ccnv 5535 Fun wfun 6352 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Homf chomf 17123 compfccomf 17124 Func cfunc 17314 Faith cfth 17364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-map 8488 df-ixp 8557 df-cat 17125 df-cid 17126 df-homf 17127 df-comf 17128 df-func 17318 df-fth 17366 |
This theorem is referenced by: (None) |
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