Theorem cofidf1 49038

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 17834	. . 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 2730	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	5, 1, 3, 6, 7, 8	cofidval 49036	. . . 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 7269	. . 3 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐹:𝐵–1-1→𝐶)
12	4, 10, 11	syl2anc 584	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)
13	2, 1, 6	funcf1 17834	. . 3 ⊢ (𝜑 → 𝐾:𝐶⟶𝐵)
14		fcofo 7270	. . 3 ⊢ ((𝐾:𝐶⟶𝐵 ∧ 𝐹:𝐵⟶𝐶 ∧ (𝐾 ∘ 𝐹) = ( I ↾ 𝐵)) → 𝐾:𝐶–onto→𝐵)
15	13, 4, 10, 14	syl3anc 1373	. 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 1540  ⟨cop 4603   class class class wbr 5115   ↦ cmpt 5196   I cid 5540   × cxp 5644   ↾ cres 5648   ∘ ccom 5650  ⟶wf 6515  –1-1→wf1 6516  –onto→wfo 6517  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396  Basecbs 17185  Hom chom 17237   Func cfunc 17822  id_funccidfu 17823   ∘_func ccofu 17824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-func 17826  df-idfu 17827  df-cofu 17828

This theorem is referenced by: (None)