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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 17905 | . 2 ⊢ Rel (𝐴 Faith 𝐶) | |
2 | relfth 17905 | . 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 17896 | . . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷)) |
12 | 11 | breqd 5163 | . . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔)) |
13 | 3 | homfeqbas 17683 | . . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵)) |
14 | 13 | raleqdv 3323 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
15 | 13, 14 | raleqbidv 3340 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
16 | 12, 15 | anbi12d 630 | . . 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 2728 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
18 | 17 | isfth 17910 | . . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦))) |
19 | eqid 2728 | . . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵) | |
20 | 19 | isfth 17910 | . . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
21 | 16, 18, 20 | 3bitr4g 313 | . 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔)) |
22 | 1, 2, 21 | eqbrrdiv 5800 | 1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 class class class wbr 5152 ◡ccnv 5681 Fun wfun 6547 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 Homf chomf 17653 compfccomf 17654 Func cfunc 17847 Faith cfth 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-ixp 8923 df-cat 17655 df-cid 17656 df-homf 17657 df-comf 17658 df-func 17851 df-fth 17901 |
This theorem is referenced by: (None) |
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