cofidf2a - Mathbox for Zhi Wang

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Theorem cofidf2a 50169

Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (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)
cofidvala.h	⊢ 𝐻 = (Hom ‘𝐷)
cofidf2a.j	⊢ 𝐽 = (Hom ‘𝐸)
cofidf2a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofidf2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
cofidf2a	⊢ (𝜑 → ((𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)) ∧ (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌)))

**Proof of Theorem cofidf2a**
Step	Hyp	Ref	Expression
1		cofidvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
2		cofidvala.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
3		cofidf2a.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
4		cofidvala.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 50128	. . . 4 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
6		cofidf2a.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		cofidf2a.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	1, 2, 3, 5, 6, 7	funcf2 18023	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)))
9		cofidvala.o	. . . . . 6 ⊢ (𝜑 → (𝐺 ∘_func 𝐹) = 𝐼)
10	9	fveq2d 6881	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺 ∘_func 𝐹)) = (2^nd ‘𝐼))
11	10	oveqd 7429	. . . 4 ⊢ (𝜑 → (𝑋(2^nd ‘(𝐺 ∘_func 𝐹))𝑌) = (𝑋(2^nd ‘𝐼)𝑌))
12		cofidvala.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))
13	1, 4, 12, 6, 7	cofu2nd 18040	. . . 4 ⊢ (𝜑 → (𝑋(2^nd ‘(𝐺 ∘_func 𝐹))𝑌) = ((((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)) ∘ (𝑋(2^nd ‘𝐹)𝑌)))
14		cofidvala.i	. . . . 5 ⊢ 𝐼 = (id_func‘𝐷)
15	5	funcrcl2 50131	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	14, 1, 15, 2, 6, 7	idfu2nd 18032	. . . 4 ⊢ (𝜑 → (𝑋(2^nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
17	11, 13, 16	3eqtr3d 2804	. . 3 ⊢ (𝜑 → ((((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)) ∘ (𝑋(2^nd ‘𝐹)𝑌)) = ( I ↾ (𝑋𝐻𝑌)))
18		fcof1 7287	. . 3 ⊢ (((𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)) ∧ ((((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)) ∘ (𝑋(2^nd ‘𝐹)𝑌)) = ( I ↾ (𝑋𝐻𝑌))) → (𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)))
19	8, 17, 18	syl2anc 596	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)))
20	14, 1, 6, 4, 12, 9	cofid1a 50164	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)) = 𝑋)
21	14, 1, 7, 4, 12, 9	cofid1a 50164	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑌)) = 𝑌)
22	20, 21	oveq12d 7430	. . . 4 ⊢ (𝜑 → (((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋))𝐻((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑌))) = (𝑋𝐻𝑌))
23		eqid 2761	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
24	12	func1st2nd 50128	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
25	1, 23, 5	funcf1 18021	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐸))
26	25, 6	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ (Base‘𝐸))
27	25, 7	ffvelcdmd 7077	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ (Base‘𝐸))
28	23, 3, 2, 24, 26, 27	funcf2 18023	. . . 4 ⊢ (𝜑 → (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))⟶(((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋))𝐻((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑌))))
29	22, 28	feq3dd 6688	. . 3 ⊢ (𝜑 → (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))⟶(𝑋𝐻𝑌))
30		fcofo 7288	. . 3 ⊢ (((((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))⟶(𝑋𝐻𝑌) ∧ (𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)) ∧ ((((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)) ∘ (𝑋(2^nd ‘𝐹)𝑌)) = ( I ↾ (𝑋𝐻𝑌))) → (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌))
31	29, 8, 17, 30	syl3anc 1398	. 2 ⊢ (𝜑 → (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌))
32	19, 31	jca 521	1 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌)) ∧ (((1^st ‘𝐹)‘𝑋)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑌)):(((1^st ‘𝐹)‘𝑋)𝐽((1^st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 I cid 5545 ↾ cres 5653 ∘ ccom 5655 ⟶wf 6527 –1-1→wf1 6528 –onto→wfo 6529 ‘cfv 6531 (class class class)co 7412 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: cofidf2 50172

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