Theorem cofidf2 50054

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 5108	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
5	3, 4	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6		cofidval.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
7		df-br 5108	. . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
8	6, 7	sylib 221	. . 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 50051	. 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 50012	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	15	oveqd 7434	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
17		eqidd 2763	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
18	3	func1st 50011	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	18	fveq1d 6884	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
20	18	fveq1d 6884	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
21	19, 20	oveq12d 7435	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
22	16, 17, 21	f1eq123d 6813	. . 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 50012	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	23, 19, 20	oveq123d 7438	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
25	24, 21, 17	foeq123d 6814	. . 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 644	. 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 235	1 ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107  –1-1→wf1 6534  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307  Hom chom 17359   Func cfunc 17949  id_funccidfu 17950   ∘_func ccofu 17951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ixp 8909  df-func 17953  df-idfu 17954  df-cofu 17955

This theorem is used by:  cofidfth  50096