cofidf1a - Mathbox for Zhi Wang

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Theorem cofidf1a 50170

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 50128	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
5	1, 2, 4	funcf1 18021	. . 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 2761	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	6, 1, 3, 7, 8, 9	cofidvala 50168	. . . 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 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 596	. 2 ⊢ (𝜑 → (1^st ‘𝐹):𝐵–1-1→𝐶)
14	7	func1st2nd 50128	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
15	2, 1, 14	funcf1 18021	. . 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 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 1570 ∈ wcel 2145 ↦ cmpt 5186 I cid 5545 × cxp 5649 ↾ cres 5653 ∘ ccom 5655 ⟶wf 6527 –1-1→wf1 6528 –onto→wfo 6529 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7988 2^nd c2nd 7989 Basecbs 17367 Hom chom 17419 Func cfunc 18009 id_funccidfu 18010 ∘_func ccofu 18011

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-map 8833 df-ixp 8910 df-func 18013 df-idfu 18014 df-cofu 18015

This theorem is used by: (None)

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