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Theorem fthf1 17875
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 17873 . . . 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 766 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12 simpr 484 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1311, 12oveq12d 7420 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
1411, 12oveq12d 7420 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1511fveq2d 6886 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑥) = (𝐹𝑋))
1612fveq2d 6886 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑦) = (𝐹𝑌))
1715, 16oveq12d 7420 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
1813, 14, 17f1eq123d 6816 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
1910, 18rspcdv 3596 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
208, 19rspcimdv 3594 . 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 1533  wcel 2098  wral 3053   class class class wbr 5139  1-1wf1 6531  cfv 6534  (class class class)co 7402  Basecbs 17149  Hom chom 17213   Func cfunc 17809   Faith cfth 17861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-ixp 8889  df-func 17813  df-fth 17863
This theorem is referenced by:  fthi  17876  ffthf1o  17877  fthoppc  17881  cofth  17893
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