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Theorem fthi 16401
 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)
fthi.r (𝜑𝑅 ∈ (𝑋𝐻𝑌))
fthi.s (𝜑𝑆 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
fthi (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))

Proof of Theorem fthi
StepHypRef Expression
1 isfth.b . . 3 𝐵 = (Base‘𝐶)
2 isfth.h . . 3 𝐻 = (Hom ‘𝐶)
3 isfth.j . . 3 𝐽 = (Hom ‘𝐷)
4 fthf1.f . . 3 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthf1.x . . 3 (𝜑𝑋𝐵)
6 fthf1.y . . 3 (𝜑𝑌𝐵)
71, 2, 3, 4, 5, 6fthf1 16400 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
8 fthi.r . 2 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
9 fthi.s . 2 (𝜑𝑆 ∈ (𝑋𝐻𝑌))
10 f1fveq 6420 . 2 (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ (𝑅 ∈ (𝑋𝐻𝑌) ∧ 𝑆 ∈ (𝑋𝐻𝑌))) → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
117, 8, 9, 10syl12anc 1316 1 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-func 16341  df-fth 16388 This theorem is referenced by:  fthsect  16408  fthmon  16410
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