Theorem cofull 17646

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 17573	. . 3 ⊢ Rel (𝐶 Func 𝐸)
2		fullfunc 17618	. . . . 5 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
3		cofull.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))
4	2, 3	sselid 3924	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		fullfunc 17618	. . . . 5 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
6		cofull.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))
7	5, 6	sselid 3924	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
8	4, 7	cofucl 17599	. . 3 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
9		1st2nd 7871	. . 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 7874	. . . . 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 2740	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2740	. . . . . . . 8 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
15		eqid 2740	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		relfull 17620	. . . . . . . . 9 ⊢ Rel (𝐷 Full 𝐸)
17	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Full 𝐸))
18		1st2ndbr 7874	. . . . . . . . 9 ⊢ ((Rel (𝐷 Full 𝐸) ∧ 𝐺 ∈ (𝐷 Full 𝐸)) → (1st ‘𝐺)(𝐷 Full 𝐸)(2nd ‘𝐺))
19	16, 17, 18	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐺)(𝐷 Full 𝐸)(2nd ‘𝐺))
20		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
21		relfunc 17573	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
22	4	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
23		1st2ndbr 7874	. . . . . . . . . . 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 17577	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
26		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
27	25, 26	ffvelrnd 6957	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28		simprr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
29	25, 28	ffvelrnd 6957	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
30	13, 14, 15, 19, 27, 29	fullfo 17624	. . . . . . 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 2740	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
32		relfull 17620	. . . . . . . . 9 ⊢ Rel (𝐶 Full 𝐷)
33	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Full 𝐷))
34		1st2ndbr 7874	. . . . . . . . 9 ⊢ ((Rel (𝐶 Full 𝐷) ∧ 𝐹 ∈ (𝐶 Full 𝐷)) → (1st ‘𝐹)(𝐶 Full 𝐷)(2nd ‘𝐹))
35	32, 33, 34	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Full 𝐷)(2nd ‘𝐹))
36	20, 15, 31, 35, 26, 28	fullfo 17624	. . . . . . 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 6699	. . . . . . 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 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)–onto→(((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦))))
39	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 𝐸))
40	20, 22, 39, 26, 28	cofu2nd 17596	. . . . . . 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 2741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
42	20, 22, 39, 26	cofu1 17595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑥)))
43	20, 22, 39, 28	cofu1 17595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘(𝐺 ∘func 𝐹))‘𝑦) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑦)))
44	42, 43	oveq12d 7287	. . . . . . 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 6706	. . . . . 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 256	. . . . 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 3117	. . . 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 17623	. . . 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 583	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹))(𝐶 Full 𝐸)(2nd ‘(𝐺 ∘func 𝐹)))
50		df-br 5080	. . 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 217	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐹)), (2nd ‘(𝐺 ∘func 𝐹))⟩ ∈ (𝐶 Full 𝐸))
52	10, 51	eqeltrd 2841	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ⟨cop 4573   class class class wbr 5079   ∘ ccom 5593  Rel wrel 5594  –onto→wfo 6429  ‘cfv 6431  (class class class)co 7269  1^st c1st 7820  2^nd c2nd 7821  Basecbs 16908  Hom chom 16969   Func cfunc 17565   ∘_func ccofu 17567   Full cful 17614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-map 8598  df-ixp 8667  df-cat 17373  df-cid 17374  df-func 17569  df-cofu 17571  df-full 17616

This theorem is referenced by:  coffth  17648