Theorem cofth 17875

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	⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
cofth.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))

Assertion

Ref	Expression
cofth	⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))

Proof of Theorem cofth

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17800	. . 3 ⊢ Rel (𝐶 Func 𝐸)
2		fthfunc 17847	. . . . 5 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
3		cofth.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))
4	2, 3	sselid 3941	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		fthfunc 17847	. . . . 5 ⊢ (𝐷 Faith 𝐸) ⊆ (𝐷 Func 𝐸)
6		cofth.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))
7	5, 6	sselid 3941	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
8	4, 7	cofucl 17826	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
9		1st2nd 7997	. . 3 ⊢ ((Rel (𝐶 Func 𝐸) ∧ (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸)) → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
10	1, 8, 9	sylancr 587	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
11		1st2ndbr 8000	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐸) ∧ (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸)) → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
12	1, 8, 11	sylancr 587	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
13		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
15		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
16		relfth 17849	. . . . . . . . 9 ⊢ Rel (𝐷 Faith 𝐸)
17	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Faith 𝐸))
18		1st2ndbr 8000	. . . . . . . . 9 ⊢ ((Rel (𝐷 Faith 𝐸) ∧ 𝐺 ∈ (𝐷 Faith 𝐸)) → (1st ‘𝐺)(𝐷 Faith 𝐸)(2nd ‘𝐺))
19	16, 17, 18	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐷 Faith 𝐸)(2nd ‘𝐺))
20		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		relfunc 17800	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
22	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
23		1st2ndbr 8000	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
24	21, 22, 23	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
25	20, 13, 24	funcf1 17804	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	ffvelcdmd 7039	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
29	25, 28	ffvelcdmd 7039	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
30	13, 14, 15, 19, 27, 29	fthf1 17857	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))–1-1→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
31		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
32		relfth 17849	. . . . . . . . 9 ⊢ Rel (𝐶 Faith 𝐷)
33	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Faith 𝐷))
34		1st2ndbr 8000	. . . . . . . . 9 ⊢ ((Rel (𝐶 Faith 𝐷) ∧ 𝐹 ∈ (𝐶 Faith 𝐷)) → (1st ‘𝐹)(𝐶 Faith 𝐷)(2nd ‘𝐹))
35	32, 33, 34	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Faith 𝐷)(2nd ‘𝐹))
36	20, 31, 14, 35, 26, 28	fthf1 17857	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
37		f1co 6749	. . . . . . 7 ⊢ (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))–1-1→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∧ (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
38	30, 36, 37	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
39	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
40	20, 22, 39, 26, 28	cofu2nd 17823	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦) = ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
41		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
42	20, 22, 39, 26	cofu1 17822	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
43	20, 22, 39, 28	cofu1 17822	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
44	42, 43	oveq12d 7387	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) = (((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
45	40, 41, 44	f1eq123d 6774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)) ↔ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))))
46	38, 45	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)))
47	46	ralrimivva 3178	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)))
48	20, 31, 15	isfth2 17855	. . . 4 ⊢ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ↔ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–1-1→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦))))
49	12, 47, 48	sylanbrc 583	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
50		df-br 5103	. . 3 ⊢ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Faith 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ↔ ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
51	49, 50	sylib 218	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Faith 𝐸))
52	10, 51	eqeltrd 2828	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4591   class class class wbr 5102   ∘ ccom 5635  Rel wrel 5636  –1-1→wf1 6496  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Hom chom 17207   Func cfunc 17792   ∘_func ccofu 17794   Faith cfth 17843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-ixp 8848  df-cat 17605  df-cid 17606  df-func 17796  df-cofu 17798  df-fth 17845

This theorem is referenced by:  coffth  17876