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Theorem fthi 16349
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 16348 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
8 fthi.r . 2 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
9 fthi.s . 2 (𝜑𝑆 ∈ (𝑋𝐻𝑌))
10 f1fveq 6397 . 2 (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ (𝑅 ∈ (𝑋𝐻𝑌) ∧ 𝑆 ∈ (𝑋𝐻𝑌))) → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
117, 8, 9, 10syl12anc 1315 1 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976   class class class wbr 4577  1-1wf1 5786  cfv 5789  (class class class)co 6526  Basecbs 15643  Hom chom 15727   Faith cfth 16334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-func 16289  df-fth 16336
This theorem is referenced by:  fthsect  16356  fthmon  16358
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