cofidf1a - Mathbox for Zhi Wang

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Theorem cofidf1a 49035

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 48993	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
5	1, 2, 4	funcf1 17834	. . 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 2730	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	6, 1, 3, 7, 8, 9	cofidvala 49033	. . . 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 494	. . 3 ⊢ (𝜑 → ((1^st ‘𝐺) ∘ (1^st ‘𝐹)) = ( I ↾ 𝐵))
12		fcof1 7269	. . 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 584	. 2 ⊢ (𝜑 → (1^st ‘𝐹):𝐵–1-1→𝐶)
14	7	func1st2nd 48993	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
15	2, 1, 14	funcf1 17834	. . 3 ⊢ (𝜑 → (1^st ‘𝐺):𝐶⟶𝐵)
16		fcofo 7270	. . 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 1373	. 2 ⊢ (𝜑 → (1^st ‘𝐺):𝐶–onto→𝐵)
18	13, 17	jca 511	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 395 = wceq 1540 ∈ wcel 2109 ↦ 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 1^st c1st 7975 2^nd c2nd 7976 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)

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