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Theorem sectmon 16363
Description: If 𝐹 is a section of 𝐺, then 𝐹 is a monomorphism. 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 (𝜑𝐶 ∈ Cat)
sectmon.x (𝜑𝑋𝐵)
sectmon.y (𝜑𝑌𝐵)
sectmon.1 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
Assertion
Ref Expression
sectmon (𝜑𝐹 ∈ (𝑋𝑀𝑌))

Proof of Theorem sectmon
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sectmon.1 . . . 4 (𝜑𝐹(𝑋𝑆𝑌)𝐺)
2 sectmon.b . . . . 5 𝐵 = (Base‘𝐶)
3 eqid 2621 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2621 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2621 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
6 sectmon.s . . . . 5 𝑆 = (Sect‘𝐶)
7 sectmon.c . . . . 5 (𝜑𝐶 ∈ Cat)
8 sectmon.x . . . . 5 (𝜑𝑋𝐵)
9 sectmon.y . . . . 5 (𝜑𝑌𝐵)
102, 3, 4, 5, 6, 7, 8, 9issect 16334 . . . 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))))
111, 10mpbid 222 . . 3 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1211simp1d 1071 . 2 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 oveq2 6612 . . . . 5 ((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)) → (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))))
1411simp3d 1073 . . . . . . . . 9 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1514ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1615oveq1d 6619 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (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))
177ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
18 simplr 791 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑥𝐵)
198ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
209ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
21 simprl 793 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋))
2212ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
2311simp2d 1072 . . . . . . . . 9 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
2423ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 16268 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (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)))
262, 3, 5, 17, 18, 4, 19, 21catlid 16265 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
2716, 25, 263eqtr3d 2663 . . . . . 6 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = 𝑔)
2815oveq1d 6619 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)))
29 simprr 795 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ∈ (𝑥(Hom ‘𝐶)𝑋))
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 16268 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌))))
312, 3, 5, 17, 18, 4, 19, 29catlid 16265 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = )
3228, 30, 313eqtr3d 2663 . . . . . 6 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑥, 𝑌⟩(comp‘𝐶)𝑋)(𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌))) = )
3327, 32eqeq12d 2636 . . . . 5 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (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))) ↔ 𝑔 = ))
3413, 33syl5ib 234 . . . 4 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → ((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
3534ralrimivva 2965 . . 3 ((𝜑𝑥𝐵) → ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
3635ralrimiva 2960 . 2 (𝜑 → ∀𝑥𝐵𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))
37 sectmon.m . . 3 𝑀 = (Mono‘𝐶)
382, 3, 4, 37, 7, 8, 9ismon2 16315 . 2 (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ ∀𝑥𝐵𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋)∀ ∈ (𝑥(Hom ‘𝐶)𝑋)((𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑌)) → 𝑔 = ))))
3912, 36, 38mpbir2and 956 1 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cop 4154   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  compcco 15874  Catccat 16246  Idccid 16247  Monocmon 16309  Sectcsect 16325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-cat 16250  df-cid 16251  df-mon 16311  df-sect 16328
This theorem is referenced by:  sectepi  16365
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