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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 5083 | . . . . 5 ⊢ (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) | |
6 | 4, 5 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ∧ 𝐹(𝐶 Faith 𝐷)𝐺)) |
7 | 6 | simprd 499 | . . 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 17295 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
11 | 6 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) |
12 | 1, 3, 2, 11, 8, 9 | fullfo 17290 | . 2 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
13 | df-f1o 6347 | . 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 586 | 1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3843 class class class wbr 5031 –1-1→wf1 6337 –onto→wfo 6338 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 Hom chom 16682 Full cful 17280 Faith cfth 17281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-map 8442 df-ixp 8511 df-func 17236 df-full 17282 df-fth 17283 |
This theorem is referenced by: catcisolem 17485 |
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