Theorem cofidf2 49245

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofidval.i	⊢ 𝐼 = (idfunc‘𝐷)
cofidval.b	⊢ 𝐵 = (Base‘𝐷)
cofidval.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
cofidval.k	⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
cofidval.o	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
cofidval.h	⊢ 𝐻 = (Hom ‘𝐷)
cofidf2.j	⊢ 𝐽 = (Hom ‘𝐸)
cofidf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofidf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
cofidf2	⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))

Proof of Theorem cofidf2

Step	Hyp	Ref	Expression
1		cofidval.i	. . 3 ⊢ 𝐼 = (idfunc‘𝐷)
2		cofidval.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
3		cofidval.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4		df-br 5094	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
5	3, 4	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6		cofidval.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
7		df-br 5094	. . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
8	6, 7	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
9		cofidval.o	. . 3 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
10		cofidval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
11		cofidf2.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
12		cofidf2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13		cofidf2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	1, 2, 5, 8, 9, 10, 11, 12, 13	cofidf2a 49242	. 2 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ∧ (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌)))
15	3	func2nd 49203	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	15	oveqd 7369	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
17		eqidd 2734	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
18	3	func1st 49202	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	18	fveq1d 6830	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
20	18	fveq1d 6830	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
21	19, 20	oveq12d 7370	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
22	16, 17, 21	f1eq123d 6760	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
23	6	func2nd 49203	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	23, 19, 20	oveq123d 7373	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
25	24, 21, 17	foeq123d 6761	. . 3 ⊢ (𝜑 → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌) ↔ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))
26	22, 25	anbi12d 632	. 2 ⊢ (𝜑 → (((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ∧ (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌)) ↔ ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌))))
27	14, 26	mpbid 232	1 ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4581   class class class wbr 5093  –1-1→wf1 6483  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7352  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17122  Hom chom 17174   Func cfunc 17763  id_funccidfu 17764   ∘_func ccofu 17765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-func 17767  df-idfu 17768  df-cofu 17769

This theorem is referenced by:  cofidfth  49287