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Mirrors > Home > MPE Home > Th. List > ffthf1o | Structured version Visualization version GIF version |
Description: The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.) |
Ref | Expression |
---|---|
isfth.b | ⊢ 𝐵 = (Base‘𝐶) |
isfth.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isfth.j | ⊢ 𝐽 = (Hom ‘𝐷) |
ffthf1o.f | ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺) |
ffthf1o.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ffthf1o.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ffthf1o | ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfth.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | isfth.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | isfth.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
4 | ffthf1o.f | . . . . 5 ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺) | |
5 | brin 5200 | . . . . 5 ⊢ (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) |
7 | 6 | simprd 495 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
8 | ffthf1o.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | ffthf1o.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 1, 2, 3, 7, 8, 9 | fthf1 17906 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
11 | 6 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | 1, 3, 2, 11, 8, 9 | fullfo 17901 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
13 | df-f1o 6555 | . 2 ⊢ ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↔ ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))) | |
14 | 10, 12, 13 | sylanbrc 582 | 1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 class class class wbr 5148 –1-1→wf1 6545 –onto→wfo 6546 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 Hom chom 17244 Full cful 17891 Faith cfth 17892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8847 df-ixp 8917 df-func 17844 df-full 17893 df-fth 17894 |
This theorem is referenced by: catcisolem 18099 |
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