MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffthf1o Structured version   Visualization version   GIF version

Theorem ffthf1o 16964
Description: The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
isfth.b 𝐵 = (Base‘𝐶)
isfth.h 𝐻 = (Hom ‘𝐶)
isfth.j 𝐽 = (Hom ‘𝐷)
ffthf1o.f (𝜑𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)
ffthf1o.x (𝜑𝑋𝐵)
ffthf1o.y (𝜑𝑌𝐵)
Assertion
Ref Expression
ffthf1o (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹𝑋)𝐽(𝐹𝑌)))

Proof of Theorem ffthf1o
StepHypRef 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 (𝜑𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)
5 brin 4938 . . . . 5 (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺))
64, 5sylib 210 . . . 4 (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺))
76simprd 491 . . 3 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
8 ffthf1o.x . . 3 (𝜑𝑋𝐵)
9 ffthf1o.y . . 3 (𝜑𝑌𝐵)
101, 2, 3, 7, 8, 9fthf1 16962 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
116simpld 490 . . 3 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
121, 3, 2, 11, 8, 9fullfo 16957 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))
13 df-f1o 6142 . 2 ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹𝑋)𝐽(𝐹𝑌)) ↔ ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌))))
1410, 12, 13sylanbrc 578 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  cin 3790   class class class wbr 4886  1-1wf1 6132  ontowfo 6133  1-1-ontowf1o 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
  Copyright terms: Public domain W3C validator