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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 4938 | . . . . 5 ⊢ (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) | |
6 | 4, 5 | sylib 210 | . . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) |
7 | 6 | simprd 491 | . . 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 16962 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
11 | 6 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | 1, 3, 2, 11, 8, 9 | fullfo 16957 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
13 | df-f1o 6142 | . 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 578 | 1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∩ cin 3790 class class class wbr 4886 –1-1→wf1 6132 –onto→wfo 6133 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Hom chom 16349 Full cful 16947 Faith cfth 16948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-map 8142 df-ixp 8195 df-func 16903 df-full 16949 df-fth 16950 |
This theorem is referenced by: catcisolem 17141 |
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