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

Theorem ffthf1o 17908
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 5200 . . . . 5 (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺))
64, 5sylib 217 . . . 4 (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Faith 𝐷)𝐺))
76simprd 495 . . 3 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
8 ffthf1o.x . . 3 (𝜑𝑋𝐵)
9 ffthf1o.y . . 3 (𝜑𝑌𝐵)
101, 2, 3, 7, 8, 9fthf1 17906 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
116simpld 494 . . 3 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
121, 3, 2, 11, 8, 9fullfo 17901 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹𝑋)𝐽(𝐹𝑌)))
13 df-f1o 6555 . 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 582 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cin 3946   class class class wbr 5148  1-1wf1 6545  ontowfo 6546  1-1-ontowf1o 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
  Copyright terms: Public domain W3C validator