Theorem cofidf1 49608

Description: If "⟨𝐹, 𝐺⟩ is a section of ⟨𝐾, 𝐿⟩ " in a category of small categories (in a universe), then 𝐹 is injective, and 𝐾 is surjective. (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)
cofidf1.c	⊢ 𝐶 = (Base‘𝐸)

Assertion

Ref	Expression
cofidf1	⊢ (𝜑 → (𝐹:𝐵–1-1→𝐶 ∧ 𝐾:𝐶–onto→𝐵))

Proof of Theorem cofidf1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofidval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		cofidf1.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
3		cofidval.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
4	1, 2, 3	funcf1 17824	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
5		cofidval.i	. . . . 5 ⊢ 𝐼 = (idfunc‘𝐷)
6		cofidval.k	. . . . 5 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)
7		cofidval.o	. . . . 5 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)
8		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	5, 1, 3, 6, 7, 8	cofidval 49606	. . . 4 ⊢ (𝜑 → ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑧)))))
10	9	simpld 494	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) = ( I ↾ 𝐵))
11		fcof1 7235	. . 3 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1→𝐶)
12	4, 10, 11	syl2anc 585	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
13	2, 1, 6	funcf1 17824	. . 3 ⊢ (𝜑 → 𝐾:𝐶⟶𝐵)
14		fcofo 7236	. . 3 ⊢ ((𝐾:𝐶⟶𝐵 ∧ 𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐾:𝐶–onto→𝐵)
15	13, 4, 10, 14	syl3anc 1374	. 2 ⊢ (𝜑 → 𝐾:𝐶–onto→𝐵)
16	12, 15	jca 511	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→𝐶 ∧ 𝐾:𝐶–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   I cid 5518   × cxp 5622   ↾ cres 5626   ∘ ccom 5628  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  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: (None)