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Theorem cofth 16359
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 16286 . . 3 Rel (𝐶 Func 𝐸)
2 fthfunc 16331 . . . . 5 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
3 cofth.f . . . . 5 (𝜑𝐹 ∈ (𝐶 Faith 𝐷))
42, 3sseldi 3560 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 fthfunc 16331 . . . . 5 (𝐷 Faith 𝐸) ⊆ (𝐷 Func 𝐸)
6 cofth.g . . . . 5 (𝜑𝐺 ∈ (𝐷 Faith 𝐸))
75, 6sseldi 3560 . . . 4 (𝜑𝐺 ∈ (𝐷 Func 𝐸))
84, 7cofucl 16312 . . 3 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Func 𝐸))
9 1st2nd 7077 . . 3 ((Rel (𝐶 Func 𝐸) ∧ (𝐺func 𝐹) ∈ (𝐶 Func 𝐸)) → (𝐺func 𝐹) = ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩)
101, 8, 9sylancr 693 . 2 (𝜑 → (𝐺func 𝐹) = ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩)
11 1st2ndbr 7080 . . . . 5 ((Rel (𝐶 Func 𝐸) ∧ (𝐺func 𝐹) ∈ (𝐶 Func 𝐸)) → (1st ‘(𝐺func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺func 𝐹)))
121, 8, 11sylancr 693 . . . 4 (𝜑 → (1st ‘(𝐺func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺func 𝐹)))
13 eqid 2604 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2604 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
15 eqid 2604 . . . . . . . 8 (Hom ‘𝐸) = (Hom ‘𝐸)
16 relfth 16333 . . . . . . . . 9 Rel (𝐷 Faith 𝐸)
176adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Faith 𝐸))
18 1st2ndbr 7080 . . . . . . . . 9 ((Rel (𝐷 Faith 𝐸) ∧ 𝐺 ∈ (𝐷 Faith 𝐸)) → (1st𝐺)(𝐷 Faith 𝐸)(2nd𝐺))
1916, 17, 18sylancr 693 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐷 Faith 𝐸)(2nd𝐺))
20 eqid 2604 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
21 relfunc 16286 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
224adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
23 1st2ndbr 7080 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2421, 22, 23sylancr 693 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2520, 13, 24funcf1 16290 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26 simprl 789 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
2725, 26ffvelrnd 6248 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
28 simprr 791 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
2925, 28ffvelrnd 6248 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
3013, 14, 15, 19, 27, 29fthf1 16341 . . . . . . 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 2604 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
32 relfth 16333 . . . . . . . . 9 Rel (𝐶 Faith 𝐷)
333adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Faith 𝐷))
34 1st2ndbr 7080 . . . . . . . . 9 ((Rel (𝐶 Faith 𝐷) ∧ 𝐹 ∈ (𝐶 Faith 𝐷)) → (1st𝐹)(𝐶 Faith 𝐷)(2nd𝐹))
3532, 33, 34sylancr 693 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Faith 𝐷)(2nd𝐹))
3620, 31, 14, 35, 26, 28fthf1 16341 . . . . . . 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 6003 . . . . . . 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 690 . . . . . 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 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
4020, 22, 39, 26, 28cofu2nd 16309 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺func 𝐹))𝑦) = ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))
41 eqidd 2605 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4220, 22, 39, 26cofu1 16308 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺func 𝐹))‘𝑥) = ((1st𝐺)‘((1st𝐹)‘𝑥)))
4320, 22, 39, 28cofu1 16308 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺func 𝐹))‘𝑦) = ((1st𝐺)‘((1st𝐹)‘𝑦)))
4442, 43oveq12d 6540 . . . . . . 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 6024 . . . . . 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 245 . . . . 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 2948 . . . 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 16339 . . . 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 694 . . 3 (𝜑 → (1st ‘(𝐺func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺func 𝐹)))
50 df-br 4573 . . 3 ((1st ‘(𝐺func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺func 𝐹)) ↔ ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
5149, 50sylib 206 . 2 (𝜑 → ⟨(1st ‘(𝐺func 𝐹)), (2nd ‘(𝐺func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
5210, 51eqeltrd 2682 1 (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2890  cop 4125   class class class wbr 4572  ccom 5027  Rel wrel 5028  1-1wf1 5782  cfv 5785  (class class class)co 6522  1st c1st 7029  2nd c2nd 7030  Basecbs 15636  Hom chom 15720   Func cfunc 16278  func ccofu 16280   Faith cfth 16327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
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 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-map 7718  df-ixp 7767  df-cat 16093  df-cid 16094  df-func 16282  df-cofu 16284  df-fth 16329
This theorem is referenced by:  coffth  16360
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