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Theorem sectmon 17054
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 2824 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2824 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2824 . . . . 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 17025 . . . 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 235 . . 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 7159 . . . . 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 725 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1615oveq1d 7166 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
18 simplr 768 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑥𝐵)
198ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
209ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
21 simprl 770 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋))
2212ad2antrr 725 . . . . . . . 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 725 . . . . . . . 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 16959 . . . . . . 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 16956 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
2716, 25, 263eqtr3d 2867 . . . . . 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 7166 . . . . . . 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 772 . . . . . . . 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 16959 . . . . . . 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 16956 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = )
3228, 30, 313eqtr3d 2867 . . . . . 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 2840 . . . . 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 247 . . . 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 3186 . . 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 3177 . 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 17006 . 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 712 1 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cop 4556   class class class wbr 5053  cfv 6345  (class class class)co 7151  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938  Monocmon 17000  Sectcsect 17016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-cat 16941  df-cid 16942  df-mon 17002  df-sect 17019
This theorem is referenced by:  sectepi  17056
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