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Theorem sectmon 17729
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 𝐡 = (Baseβ€˜πΆ)
sectmon.m 𝑀 = (Monoβ€˜πΆ)
sectmon.s 𝑆 = (Sectβ€˜πΆ)
sectmon.c (πœ‘ β†’ 𝐢 ∈ Cat)
sectmon.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
sectmon.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
sectmon.1 (πœ‘ β†’ 𝐹(π‘‹π‘†π‘Œ)𝐺)
Assertion
Ref Expression
sectmon (πœ‘ β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))

Proof of Theorem sectmon
Dummy variables 𝑔 β„Ž π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sectmon.1 . . . 4 (πœ‘ β†’ 𝐹(π‘‹π‘†π‘Œ)𝐺)
2 sectmon.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
3 eqid 2733 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
4 eqid 2733 . . . . 5 (compβ€˜πΆ) = (compβ€˜πΆ)
5 eqid 2733 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
6 sectmon.s . . . . 5 𝑆 = (Sectβ€˜πΆ)
7 sectmon.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
8 sectmon.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
9 sectmon.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝐡)
102, 3, 4, 5, 6, 7, 8, 9issect 17700 . . . 4 (πœ‘ β†’ (𝐹(π‘‹π‘†π‘Œ)𝐺 ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ 𝐺 ∈ (π‘Œ(Hom β€˜πΆ)𝑋) ∧ (𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ((Idβ€˜πΆ)β€˜π‘‹))))
111, 10mpbid 231 . . 3 (πœ‘ β†’ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ 𝐺 ∈ (π‘Œ(Hom β€˜πΆ)𝑋) ∧ (𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ((Idβ€˜πΆ)β€˜π‘‹)))
1211simp1d 1143 . 2 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
13 oveq2 7417 . . . . 5 ((𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) = (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž)))
1411simp3d 1145 . . . . . . . . 9 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ((Idβ€˜πΆ)β€˜π‘‹))
1514ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹) = ((Idβ€˜πΆ)β€˜π‘‹))
1615oveq1d 7424 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝑔) = (((Idβ€˜πΆ)β€˜π‘‹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝑔))
177ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ 𝐢 ∈ Cat)
18 simplr 768 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ π‘₯ ∈ 𝐡)
198ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ 𝑋 ∈ 𝐡)
209ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ π‘Œ ∈ 𝐡)
21 simprl 770 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋))
2212ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
2311simp2d 1144 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
2423ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ 𝐺 ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 17630 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝑔) = (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)))
262, 3, 5, 17, 18, 4, 19, 21catlid 17627 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ (((Idβ€˜πΆ)β€˜π‘‹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)𝑔) = 𝑔)
2716, 25, 263eqtr3d 2781 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) = 𝑔)
2815oveq1d 7424 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)β„Ž) = (((Idβ€˜πΆ)β€˜π‘‹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)β„Ž))
29 simprr 772 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 17630 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)𝑋)𝐹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)β„Ž) = (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž)))
312, 3, 5, 17, 18, 4, 19, 29catlid 17627 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ (((Idβ€˜πΆ)β€˜π‘‹)(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)𝑋)β„Ž) = β„Ž)
3228, 30, 313eqtr3d 2781 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž)) = β„Ž)
3327, 32eqeq12d 2749 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) = (𝐺(⟨π‘₯, π‘ŒβŸ©(compβ€˜πΆ)𝑋)(𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž)) ↔ 𝑔 = β„Ž))
3413, 33imbitrid 243 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝐡) ∧ (𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑋) ∧ β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋))) β†’ ((𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
3534ralrimivva 3201 . . 3 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ βˆ€π‘” ∈ (π‘₯(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋)((𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
3635ralrimiva 3147 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘” ∈ (π‘₯(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋)((𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
37 sectmon.m . . 3 𝑀 = (Monoβ€˜πΆ)
382, 3, 4, 37, 7, 8, 9ismon2 17681 . 2 (πœ‘ β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ (𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘” ∈ (π‘₯(Hom β€˜πΆ)𝑋)βˆ€β„Ž ∈ (π‘₯(Hom β€˜πΆ)𝑋)((𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) = (𝐹(⟨π‘₯, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))))
3912, 36, 38mpbir2and 712 1 (πœ‘ β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609  Monocmon 17675  Sectcsect 17691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cat 17612  df-cid 17613  df-mon 17677  df-sect 17694
This theorem is referenced by:  sectepi  17731
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