Theorem sectmon 17986

Description: If 𝐹 is a section of 𝐺, then 𝐹 is a monomorphism. Proposition 7.35 of [Adamek] p. 110. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
sectmon.b	⊢ 𝐵 = (Base‘𝐶)
sectmon.m	⊢ 𝑀 = (Mono‘𝐶)
sectmon.s	⊢ 𝑆 = (Sect‘𝐶)
sectmon.c	⊢ (𝜑 → 𝐶 ∈ Cat)
sectmon.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
sectmon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
sectmon.1	⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)

Assertion

Ref	Expression
sectmon	⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))

Proof of Theorem sectmon

Dummy variables 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sectmon.1	. . . 4 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
2		sectmon.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		eqid 2763	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		sectmon.s	. . . . 5 ⊢ 𝑆 = (Sect‘𝐶)
7		sectmon.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		sectmon.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		sectmon.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	2, 3, 4, 5, 6, 7, 8, 9	issect 17957	. . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
11	1, 10	mpbid 235	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
12	11	simp1d 1160	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
13		oveq2 7451	. . . . 5 ⊢ ((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ)))
14	11	simp3d 1162	. . . . . . . . 9 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
15	14	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
16	15	oveq1d 7458	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔))
17	7	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
18		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑥 ∈ 𝐵)
19	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑋 ∈ 𝐵)
20	9	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
21		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋))
22	12	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
23	11	simp2d 1161	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
24	23	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
25	2, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24	catass 17889	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
26	2, 3, 5, 17, 18, 4, 19, 21	catlid 17886	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
27	16, 25, 26	3eqtr3d 2806	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = 𝑔)
28	15	oveq1d 7458	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)ℎ))
29		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))
30	2, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24	catass 17889	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ)))
31	2, 3, 5, 17, 18, 4, 19, 29	catlid 17886	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = ℎ)
32	28, 30, 31	3eqtr3d 2806	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ)) = ℎ)
33	27, 32	eqeq12d 2779	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ)) ↔ 𝑔 = ℎ))
34	13, 33	imbitrid 247	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ℎ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
35	34	ralrimivva 3209	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
36	35	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
37		sectmon.m	. . 3 ⊢ 𝑀 = (Mono‘𝐶)
38	2, 3, 4, 37, 7, 8, 9	ismon2 17938	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑥 ∈ 𝐵 ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))))
39	12, 36, 38	mpbir2and 726	1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868  Monocmon 17932  Sectcsect 17948

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-cat 17871  df-cid 17872  df-mon 17934  df-sect 17951

This theorem is used by:  sectepi  17988