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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 17859 | . 2 ⊢ Rel (𝐴 Faith 𝐶) | |
2 | relfth 17859 | . 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 17850 | . . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷)) |
12 | 11 | breqd 5159 | . . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔)) |
13 | 3 | homfeqbas 17639 | . . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵)) |
14 | 13 | raleqdv 3325 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
15 | 13, 14 | raleqbidv 3342 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
16 | 12, 15 | anbi12d 631 | . . 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 2732 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
18 | 17 | isfth 17864 | . . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦))) |
19 | eqid 2732 | . . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵) | |
20 | 19 | isfth 17864 | . . 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 5794 | 1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 class class class wbr 5148 ◡ccnv 5675 Fun wfun 6537 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 Homf chomf 17609 compfccomf 17610 Func cfunc 17803 Faith cfth 17853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-ixp 8891 df-cat 17611 df-cid 17612 df-homf 17613 df-comf 17614 df-func 17807 df-fth 17855 |
This theorem is referenced by: (None) |
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