cofidf1a - Mathbox for Zhi Wang

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Theorem cofidf1a 49873

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 49831	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
5	1, 2, 4	funcf1 17924	. . 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 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	6, 1, 3, 7, 8, 9	cofidvala 49871	. . . 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 499	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵))
12		fcof1 7287	. . 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 595	. 2 ⊢ (𝜑 → (1^st ‘𝐹):𝐵–1-1→𝐶)
14	7	func1st2nd 49831	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
15	2, 1, 14	funcf1 17924	. . 3 ⊢ (𝜑 → (1^st ‘𝐺):𝐶⟶𝐵)
16		fcofo 7288	. . 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 520	1 ⊢ (𝜑 → ((1^st ‘𝐹):𝐵–1-1→𝐶 ∧ (1^st ‘𝐺):𝐶–onto→𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ↦ cmpt 5193 I cid 5557 × cxp 5661 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6534 –1-1→wf1 6535 –onto→wfo 6536 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 Hom chom 17322 Func cfunc 17912 id_funccidfu 17913 ∘_func ccofu 17914

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-map 8827 df-ixp 8897 df-func 17916 df-idfu 17917 df-cofu 17918

This theorem is referenced by: (None)

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