Theorem cofidf1 50331

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 18047	. . 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 2760	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
9	5, 1, 3, 6, 7, 8	cofidval 50329	. . . 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 18047	. . 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 1569  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   × cxp 5648   ↾ cres 5652   ∘ ccom 5654  ⟶wf 6535  –1-1→wf1 6536  –onto→wfo 6537  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  Basecbs 17393  Hom chom 17445   Func cfunc 18035  id_funccidfu 18036   ∘_func ccofu 18037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8853  df-ixp 8930  df-func 18039  df-idfu 18040  df-cofu 18041

This theorem is used by: (None)