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Theorem fthf1 17905
Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfth.b 𝐵 = (Base‘𝐶)
isfth.h 𝐻 = (Hom ‘𝐶)
isfth.j 𝐽 = (Hom ‘𝐷)
fthf1.f (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
fthf1.x (𝜑𝑋𝐵)
fthf1.y (𝜑𝑌𝐵)
Assertion
Ref Expression
fthf1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))

Proof of Theorem fthf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthf1.f . . 3 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
2 isfth.b . . . . 5 𝐵 = (Base‘𝐶)
3 isfth.h . . . . 5 𝐻 = (Hom ‘𝐶)
4 isfth.j . . . . 5 𝐽 = (Hom ‘𝐷)
52, 3, 4isfth2 17903 . . . 4 (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
65simprbi 496 . . 3 (𝐹(𝐶 Faith 𝐷)𝐺 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))
71, 6syl 17 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)))
8 fthf1.x . . 3 (𝜑𝑋𝐵)
9 fthf1.y . . . . 5 (𝜑𝑌𝐵)
109adantr 480 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
11 simplr 768 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12 simpr 484 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1311, 12oveq12d 7438 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
1411, 12oveq12d 7438 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1511fveq2d 6901 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑥) = (𝐹𝑋))
1612fveq2d 6901 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑦) = (𝐹𝑌))
1715, 16oveq12d 7438 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
1813, 14, 17f1eq123d 6831 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
1910, 18rspcdv 3601 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
208, 19rspcimdv 3599 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
217, 20mpd 15 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058   class class class wbr 5148  1-1wf1 6545  cfv 6548  (class class class)co 7420  Basecbs 17179  Hom chom 17243   Func cfunc 17839   Faith cfth 17891
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-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8846  df-ixp 8916  df-func 17843  df-fth 17893
This theorem is referenced by:  fthi  17906  ffthf1o  17907  fthoppc  17911  cofth  17923
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