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Theorem cofth 17827
Description: The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
cofth.f (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
cofth.g (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
Assertion
Ref Expression
cofth (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))

Proof of Theorem cofth
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 17753 . . 3 Rel (𝐶 Func 𝐸)
2 fthfunc 17799 . . . . 5 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
3 cofth.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
42, 3sselid 3943 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 fthfunc 17799 . . . . 5 (𝐷 Faith 𝐸) ⊆ (𝐷 Func 𝐸)
6 cofth.g . . . . 5 (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
75, 6sselid 3943 . . . 4 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
84, 7cofucl 17779 . . 3 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Func 𝐸))
9 1st2nd 7972 . . 3 ((Rel (𝐶 Func 𝐸) ∧ (𝐺func 𝐹) ∈ (𝐶 Func 𝐸)) → (𝐺func 𝐹) = ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩)
101, 8, 9sylancr 588 . 2 (𝜑 → (𝐺func 𝐹) = ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩)
11 1st2ndbr 7975 . . . . 5 ((Rel (𝐶 Func 𝐸) ∧ (𝐺func 𝐹) ∈ (𝐶 Func 𝐸)) → (1st ‘(𝐺func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺func 𝐹)))
121, 8, 11sylancr 588 . . . 4 (𝜑 → (1st ‘(𝐺func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺func 𝐹)))
13 eqid 2733 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2733 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
15 eqid 2733 . . . . . . . 8 (Hom ‘𝐸) = (Hom ‘𝐸)
16 relfth 17801 . . . . . . . . 9 Rel (𝐷 Faith 𝐸)
176adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Faith 𝐸))
18 1st2ndbr 7975 . . . . . . . . 9 ((Rel (𝐷 Faith 𝐸) ∧ 𝐺 ∈ (𝐷 Faith 𝐸)) → (1st𝐺)(𝐷 Faith 𝐸)(2nd𝐺))
1916, 17, 18sylancr 588 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐷 Faith 𝐸)(2nd𝐺))
20 eqid 2733 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
21 relfunc 17753 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
224adantr 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
23 1st2ndbr 7975 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2421, 22, 23sylancr 588 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2520, 13, 24funcf1 17757 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
2725, 26ffvelcdmd 7037 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
28 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
2925, 28ffvelcdmd 7037 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
3013, 14, 15, 19, 27, 29fthf1 17809 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))–1-1→(((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦))))
31 eqid 2733 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
32 relfth 17801 . . . . . . . . 9 Rel (𝐶 Faith 𝐷)
333adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Faith 𝐷))
34 1st2ndbr 7975 . . . . . . . . 9 ((Rel (𝐶 Faith 𝐷) ∧ 𝐹 ∈ (𝐶 Faith 𝐷)) → (1st𝐹)(𝐶 Faith 𝐷)(2nd𝐹))
3532, 33, 34sylancr 588 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Faith 𝐷)(2nd𝐹))
3620, 31, 14, 35, 26, 28fthf1 17809 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
37 f1co 6751 . . . . . . 7 (((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)):(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))–1-1→(((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦))) ∧ (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦))) → ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦))))
3830, 36, 37syl2anc 585 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦))))
397adantr 482 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
4020, 22, 39, 26, 28cofu2nd 17776 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺func 𝐹))𝑦) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
41 eqidd 2734 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4220, 22, 39, 26cofu1 17775 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺func 𝐹))‘𝑥) = ((1st𝐺)‘((1st𝐹)‘𝑥)))
4320, 22, 39, 28cofu1 17775 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺func 𝐹))‘𝑦) = ((1st𝐺)‘((1st𝐹)‘𝑦)))
4442, 43oveq12d 7376 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘(𝐺func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺func 𝐹))‘𝑦)) = (((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦))))
4540, 41, 44f1eq123d 6777 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd ‘(𝐺func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺func 𝐹))‘𝑦)) ↔ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st𝐺)‘((1st𝐹)‘𝑥))(Hom ‘𝐸)((1st𝐺)‘((1st𝐹)‘𝑦)))))
4638, 45mpbird 257 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺func 𝐹))‘𝑦)))
4746ralrimivva 3194 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺func 𝐹))‘𝑦)))
4820, 31, 15isfth2 17807 . . . 4 ((1st ‘(𝐺func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺func 𝐹)) ↔ ((1st ‘(𝐺func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺func 𝐹)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺func 𝐹))‘𝑦))))
4912, 47, 48sylanbrc 584 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺func 𝐹)))
50 df-br 5107 . . 3 ((1st ‘(𝐺func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺func 𝐹)) ↔ ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
5149, 50sylib 217 . 2 (𝜑 → ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
5210, 51eqeltrd 2834 1 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  cop 4593   class class class wbr 5106  ccom 5638  Rel wrel 5639  1-1wf1 6494  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Basecbs 17088  Hom chom 17149   Func cfunc 17745  func ccofu 17747   Faith cfth 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-ixp 8839  df-cat 17553  df-cid 17554  df-func 17749  df-cofu 17751  df-fth 17797
This theorem is referenced by:  coffth  17828
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