Theorem cofidf2 49607

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 5087	. . . 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 5087	. . . 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 49604	. 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 49565	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	15	oveqd 7377	. . . 4 ⊢ (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
17		eqidd 2738	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
18	3	func1st 49564	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
19	18	fveq1d 6836	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
20	18	fveq1d 6836	. . . . 5 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹‘𝑌))
21	19, 20	oveq12d 7378	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
22	16, 17, 21	f1eq123d 6766	. . 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 49565	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
24	23, 19, 20	oveq123d 7381	. . . 4 ⊢ (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹‘𝑋)𝐿(𝐹‘𝑌)))
25	24, 21, 17	foeq123d 6767	. . 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 633	. 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 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  –1-1→wf1 6489  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222   Func cfunc 17812  id_funccidfu 17813   ∘_func ccofu 17814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-ixp 8839  df-func 17816  df-idfu 17817  df-cofu 17818

This theorem is referenced by:  cofidfth  49649