cofidf1a - Mathbox for Zhi Wang

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Theorem cofidf1a 50327

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the object part of 𝐹 is injective, and the object part of 𝐺 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.)

**Hypotheses**
Ref	Expression
cofidvala.i	⊢ 𝐼 = (id_func‘𝐷)
cofidvala.b	⊢ 𝐵 = (Base‘𝐷)
cofidvala.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofidvala.g	⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
cofidvala.o	⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐼)
cofidf1a.c	⊢ 𝐶 = (Base‘𝐸)

**Assertion**
Ref	Expression
cofidf1a	⊢ (𝜑 → ((1^st ‘𝐹):𝐵–1-1→𝐶 ∧ (1^st ‘𝐺):𝐶–onto→𝐵))

Proof of Theorem cofidf1a Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofidvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		cofidf1a.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
3		cofidvala.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
4	3	func1st2nd 50283	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
5	1, 2, 4	funcf1 18046	. . 3 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐶)
6		cofidvala.i	. . . . 5 ⊢ 𝐼 = (id_func‘𝐷)
7		cofidvala.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
8		cofidvala.o	. . . . 5 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐼)
9		eqid 2760	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	6, 1, 3, 7, 8, 9	cofidvala 50325	. . . 4 ⊢ (𝜑 → (((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑦)) ∘ (𝑥(2^nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ ((Hom ‘𝐷)‘𝑧)))))
11	10	simpld 500	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵))
12		fcof1 7295	. . 3 ⊢ (((1^st ‘𝐹):𝐵⟶𝐶 ∧ ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵)) → (1^st ‘𝐹):𝐵–1-1→𝐶)
13	5, 11, 12	syl2anc 596	. 2 ⊢ (𝜑 → (1^st ‘𝐹):𝐵–1-1→𝐶)
14	7	func1st2nd 50283	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
15	2, 1, 14	funcf1 18046	. . 3 ⊢ (𝜑 → (1^st ‘𝐺):𝐶⟶𝐵)
16		fcofo 7296	. . 3 ⊢ (((1^st ‘𝐺):𝐶⟶𝐵 ∧ (1^st ‘𝐹):𝐵⟶𝐶 ∧ ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵)) → (1^st ‘𝐺):𝐶–onto→𝐵)
17	15, 5, 11, 16	syl3anc 1398	. 2 ⊢ (𝜑 → (1^st ‘𝐺):𝐶–onto→𝐵)
18	13, 17	jca 521	1 ⊢ (𝜑 → ((1^st ‘𝐹):𝐵–1-1→𝐶 ∧ (1^st ‘𝐺):𝐶–onto→𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ↦ cmpt 5185 I cid 5544 × cxp 5648 ↾ cres 5652 ∘ ccom 5654 ⟶wf 6534 –1-1→wf1 6535 –onto→wfo 6536 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 Basecbs 17392 Hom chom 17444 Func cfunc 18034 id_funccidfu 18035 ∘_func ccofu 18036

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 7751

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 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-map 8852 df-ixp 8929 df-func 18038 df-idfu 18039 df-cofu 18040

This theorem is used by: (None)

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