cofidf1a - Mathbox for Zhi Wang

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Theorem cofidf1a 49243

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 49201	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
5	1, 2, 4	funcf1 17775	. . 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 2733	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	6, 1, 3, 7, 8, 9	cofidvala 49241	. . . 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 7227	. . 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 49201	. . . 4 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
15	2, 1, 14	funcf1 17775	. . 3 ⊢ (𝜑 → (1^st ‘𝐺):𝐶⟶𝐵)
16		fcofo 7228	. . 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 1541 ∈ wcel 2113 ↦ cmpt 5174 I cid 5513 × cxp 5617 ↾ cres 5621 ∘ ccom 5623 ⟶wf 6482 –1-1→wf1 6483 –onto→wfo 6484 ‘cfv 6486 (class class class)co 7352 ∈ cmpo 7354 1^st c1st 7925 2^nd c2nd 7926 Basecbs 17122 Hom chom 17174 Func cfunc 17763 id_funccidfu 17764 ∘_func ccofu 17765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-map 8758 df-ixp 8828 df-func 17767 df-idfu 17768 df-cofu 17769

This theorem is referenced by: (None)

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