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Theorem sectmon 16827
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 2778 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2778 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
5 eqid 2778 . . . . 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 16798 . . . 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 224 . . 3 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
1211simp1d 1133 . 2 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 oveq2 6930 . . . . 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 1135 . . . . . . . . 9 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1514ad2antrr 716 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
1615oveq1d 6937 . . . . . . 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 716 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
18 simplr 759 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑥𝐵)
198ad2antrr 716 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑋𝐵)
209ad2antrr 716 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑌𝐵)
21 simprl 761 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋))
2212ad2antrr 716 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
2311simp2d 1134 . . . . . . . . 9 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))
2423ad2antrr 716 . . . . . . . 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 16732 . . . . . . 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 16729 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
2716, 25, 263eqtr3d 2822 . . . . . 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 6937 . . . . . . 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 763 . . . . . . . 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 16732 . . . . . . 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 16729 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑔 ∈ (𝑥(Hom ‘𝐶)𝑋) ∧ ∈ (𝑥(Hom ‘𝐶)𝑋))) → (((Id‘𝐶)‘𝑋)(⟨𝑥, 𝑋⟩(comp‘𝐶)𝑋)) = )
3228, 30, 313eqtr3d 2822 . . . . . 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 2793 . . . . 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 236 . . . 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 3153 . . 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 3148 . 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 16779 . 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 703 1 (𝜑𝐹 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  cop 4404   class class class wbr 4886  cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349  compcco 16350  Catccat 16710  Idccid 16711  Monocmon 16773  Sectcsect 16789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-cat 16714  df-cid 16715  df-mon 16775  df-sect 16792
This theorem is referenced by:  sectepi  16829
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