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Theorem sectmon 17768
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 2725 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2725 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2725 . . . . 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 17739 . . . 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 231 . . 3 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1211simp1d 1139 . 2 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 oveq2 7427 . . . . 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 1141 . . . . . . . . 9 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1514ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1615oveq1d 7434 . . . . . . 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 724 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
18 simplr 767 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑥𝐵)
198ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
209ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
21 simprl 769 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋))
2212ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
2311simp2d 1140 . . . . . . . . 9 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
2423ad2antrr 724 . . . . . . . 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 17669 . . . . . . 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 17666 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
2716, 25, 263eqtr3d 2773 . . . . . 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 7434 . . . . . . 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 771 . . . . . . . 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 17669 . . . . . . 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 17666 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = )
3228, 30, 313eqtr3d 2773 . . . . . 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 2741 . . . . 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, 33imbitrid 243 . . . 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 3190 . . 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 3135 . 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 17720 . 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 711 1 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cop 4636   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  compcco 17248  Catccat 17647  Idccid 17648  Monocmon 17714  Sectcsect 17730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-cat 17651  df-cid 17652  df-mon 17716  df-sect 17733
This theorem is referenced by:  sectepi  17770
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