cofidf2a - Mathbox for Zhi Wang

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Theorem cofidf2a 49702

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 49661	. . . 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 17884	. . 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 6867	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝐺 ∘_func 𝐹)) = (2^nd ‘𝐼))
11	10	oveqd 7409	. . . 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 17901	. . . 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 49664	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
16	14, 1, 15, 2, 6, 7	idfu2nd 17893	. . . 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 7267	. . 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 593	. 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 49697	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑋)) = 𝑋)
21	14, 1, 7, 4, 12, 9	cofid1a 49697	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑌)) = 𝑌)
22	20, 21	oveq12d 7410	. . . 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 49661	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺)(𝐸 Func 𝐷)(2^nd ‘𝐺))
25	1, 23, 5	funcf1 17882	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐸))
26	25, 6	ffvelcdmd 7062	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ (Base‘𝐸))
27	25, 7	ffvelcdmd 7062	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ (Base‘𝐸))
28	23, 3, 2, 24, 26, 27	funcf2 17884	. . . 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 6674	. . 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 7268	. . 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 1389	. 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 519	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

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 I cid 5539 ↾ cres 5647 ∘ ccom 5649 ⟶wf 6513 –1-1→wf1 6514 –onto→wfo 6515 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 2^nd c2nd 7965 Basecbs 17228 Hom chom 17280 Func cfunc 17870 id_funccidfu 17871 ∘_func ccofu 17872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-map 8805 df-ixp 8876 df-func 17874 df-idfu 17875 df-cofu 17876

This theorem is referenced by: cofidf2 49705

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