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Theorem fthi 17176
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 17175 . 2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)))
8 fthi.r . 2 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
9 fthi.s . 2 (𝜑𝑆 ∈ (𝑋𝐻𝑌))
10 f1fveq 7011 . 2 (((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ (𝑅 ∈ (𝑋𝐻𝑌) ∧ 𝑆 ∈ (𝑋𝐻𝑌))) → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
117, 8, 9, 10syl12anc 832 1 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105   class class class wbr 5057  1-1wf1 6345  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564   Faith cfth 17161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-ixp 8450  df-func 17116  df-fth 17163
This theorem is referenced by:  fthsect  17183  fthmon  17185
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