Theorem cofull 16801

Description: The composition of two full functors is full. Proposition 3.30(d) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
cofull.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
cofull.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))

Assertion

Ref	Expression
cofull	⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))

Proof of Theorem cofull

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

Step	Hyp	Ref	Expression
1		relfunc 16729	. . 3 ⊢ Rel (𝐶 Func 𝐸)
2		fullfunc 16773	. . . . 5 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
3		cofull.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
4	2, 3	sseldi 3803	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		fullfunc 16773	. . . . 5 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
6		cofull.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
7	5, 6	sseldi 3803	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
8	4, 7	cofucl 16755	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
9		1st2nd 7449	. . 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 577	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩)
11		1st2ndbr 7452	. . . . 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 577	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Func 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
13		eqid 2813	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2813	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
15		eqid 2813	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		relfull 16775	. . . . . . . . 9 ⊢ Rel (𝐷 Full 𝐸)
17	6	adantr 468	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Full 𝐸))
18		1st2ndbr 7452	. . . . . . . . 9 ⊢ ((Rel (𝐷 Full 𝐸) ∧ 𝐺 ∈ (𝐷 Full 𝐸)) → (1st ‘𝐺)(𝐷 Full 𝐸)(2nd ‘𝐺))
19	16, 17, 18	sylancr 577	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐷 Full 𝐸)(2nd ‘𝐺))
20		eqid 2813	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		relfunc 16729	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
22	4	adantr 468	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
23		1st2ndbr 7452	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
24	21, 22, 23	sylancr 577	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
25	20, 13, 24	funcf1 16733	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		simprl 778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	ffvelrnd 6585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28		simprr 780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
29	25, 28	ffvelrnd 6585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
30	13, 14, 15, 19, 27, 29	fullfo 16779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
31		eqid 2813	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
32		relfull 16775	. . . . . . . . 9 ⊢ Rel (𝐶 Full 𝐷)
33	3	adantr 468	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Full 𝐷))
34		1st2ndbr 7452	. . . . . . . . 9 ⊢ ((Rel (𝐶 Full 𝐷) ∧ 𝐹 ∈ (𝐶 Full 𝐷)) → (1st ‘𝐹)(𝐶 Full 𝐷)(2nd ‘𝐹))
35	32, 33, 34	sylancr 577	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Full 𝐷)(2nd ‘𝐹))
36	20, 15, 31, 35, 26, 28	fullfo 16779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
37		foco 6344	. . . . . . 7 ⊢ (((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)):(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))) ∧ (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
38	30, 36, 37	syl2anc 575	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
39	7	adantr 468	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
40	20, 22, 39, 26, 28	cofu2nd 16752	. . . . . . 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 2814	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
42	20, 22, 39, 26	cofu1 16751	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
43	20, 22, 39, 28	cofu1 16751	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
44	42, 43	oveq12d 6895	. . . . . . 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	foeq123d 6351	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–onto→(((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)–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))))
46	38, 45	mpbird 248	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)))
47	46	ralrimivva 3166	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(2nd ‘(𝐺 ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐶)𝑦)–onto→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦)))
48	20, 14, 31	isfull2 16778	. . . 4 ⊢ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Full 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)–onto→(((1st ‘(𝐺 ∘func 𝐹))‘𝑥)(Hom ‘𝐸)((1st ‘(𝐺 ∘func 𝐹))‘𝑦))))
49	12, 47, 48	sylanbrc 574	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Full 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
50		df-br 4852	. . 3 ⊢ ((1st ‘(𝐺 ∘func 𝐹))(𝐶 Full 𝐸)(2nd ‘(𝐺 ∘func 𝐹)) ↔ ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Full 𝐸))
51	49, 50	sylib 209	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Full 𝐸))
52	10, 51	eqeltrd 2892	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2157  ∀wral 3103  ⟨cop 4383   class class class wbr 4851   ∘ ccom 5322  Rel wrel 5323  –onto→wfo 6102  ‘cfv 6104  (class class class)co 6877  1^st c1st 7399  2^nd c2nd 7400  Basecbs 16071  Hom chom 16167   Func cfunc 16721   ∘_func ccofu 16723   Full cful 16769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1st 7401  df-2nd 7402  df-map 8097  df-ixp 8149  df-cat 16536  df-cid 16537  df-func 16725  df-cofu 16727  df-full 16771

This theorem is referenced by:  coffth  16803