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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 17801 | . 2 ⊢ Rel (𝐴 Faith 𝐶) | |
2 | relfth 17801 | . 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 17792 | . . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷)) |
12 | 11 | breqd 5117 | . . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔)) |
13 | 3 | homfeqbas 17581 | . . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵)) |
14 | 13 | raleqdv 3312 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
15 | 13, 14 | raleqbidv 3318 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
16 | 12, 15 | anbi12d 632 | . . 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 2733 | . . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
18 | 17 | isfth 17806 | . . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦))) |
19 | eqid 2733 | . . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵) | |
20 | 19 | isfth 17806 | . . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))) |
21 | 16, 18, 20 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔)) |
22 | 1, 2, 21 | eqbrrdiv 5751 | 1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 class class class wbr 5106 ◡ccnv 5633 Fun wfun 6491 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 Homf chomf 17551 compfccomf 17552 Func cfunc 17745 Faith cfth 17795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-ixp 8839 df-cat 17553 df-cid 17554 df-homf 17555 df-comf 17556 df-func 17749 df-fth 17797 |
This theorem is referenced by: (None) |
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