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Theorem fthf1 16962
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 16960 . . . 4 (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦))))
65simprbi 492 . . 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 474 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
11 simplr 759 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12 simpr 479 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
1311, 12oveq12d 6940 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
1411, 12oveq12d 6940 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1511fveq2d 6450 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑥) = (𝐹𝑋))
1612fveq2d 6450 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (𝐹𝑦) = (𝐹𝑌))
1715, 16oveq12d 6940 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝐹𝑥)𝐽(𝐹𝑦)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
1813, 14, 17f1eq123d 6384 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
1910, 18rspcdv 3514 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹𝑥)𝐽(𝐹𝑦)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
208, 19rspcimdv 3512 . 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 386   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4886  1-1wf1 6132  cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349   Func cfunc 16899   Faith cfth 16948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-ixp 8195  df-func 16903  df-fth 16950
This theorem is referenced by:  fthi  16963  ffthf1o  16964  fthoppc  16968  cofth  16980
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