Theorem cofidf2 49881

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 5111	. . . 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 5111	. . . 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 49878	. 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 49839	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	15	oveqd 7429	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
17		eqidd 2764	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
18	3	func1st 49838	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	18	fveq1d 6885	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
20	18	fveq1d 6885	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
21	19, 20	oveq12d 7430	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
22	16, 17, 21	f1eq123d 6814	. . 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 49839	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	23, 19, 20	oveq123d 7433	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
25	24, 21, 17	foeq123d 6815	. . 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 643	. 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)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4596   class class class wbr 5110  –1-1→wf1 6535  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322   Func cfunc 17912  id_funccidfu 17913   ∘_func ccofu 17914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-ixp 8897  df-func 17916  df-idfu 17917  df-cofu 17918

This theorem is referenced by:  cofidfth  49923