Theorem cofidf1 49940

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 17948	. . 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 2766	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	5, 1, 3, 6, 7, 8	cofidval 49938	. . . 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 500	. . 3 ⊢ (𝜑 → (𝐾 ∘ 𝐹) = ( I ↾ 𝐵))
11		fcof1 7296	. . 3 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1→𝐶)
12	4, 10, 11	syl2anc 596	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
13	2, 1, 6	funcf1 17948	. . 3 ⊢ (𝜑 → 𝐾:𝐶⟶𝐵)
14		fcofo 7297	. . 3 ⊢ ((𝐾:𝐶⟶𝐵 ∧ 𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐾:𝐶–onto→𝐵)
15	13, 4, 10, 14	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐾:𝐶–onto→𝐵)
16	12, 15	jca 521	1 ⊢ (𝜑 → (𝐹:𝐵–1-1→𝐶 ∧ 𝐾:𝐶–onto→𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ⟨cop 4600   class class class wbr 5114   ↦ cmpt 5197   I cid 5560   × cxp 5664   ↾ cres 5668   ∘ ccom 5670  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  Basecbs 17294  Hom chom 17346   Func cfunc 17936  id_funccidfu 17937   ∘_func ccofu 17938

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-map 8835  df-ixp 8905  df-func 17940  df-idfu 17941  df-cofu 17942

This theorem is used by: (None)