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Theorem monfval 17575
Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
ismon.b 𝐡 = (Baseβ€˜πΆ)
ismon.h 𝐻 = (Hom β€˜πΆ)
ismon.o Β· = (compβ€˜πΆ)
ismon.s 𝑀 = (Monoβ€˜πΆ)
ismon.c (πœ‘ β†’ 𝐢 ∈ Cat)
Assertion
Ref Expression
monfval (πœ‘ β†’ 𝑀 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝑦,𝑧,𝐡   πœ‘,𝑓,𝑔,π‘₯,𝑦,𝑧   𝐢,𝑓,𝑔,π‘₯,𝑦,𝑧   𝑓,𝐻,𝑔,π‘₯,𝑦,𝑧   Β· ,𝑓,𝑔,π‘₯,𝑦,𝑧   𝑓,𝑀
Allowed substitution hints:   𝑀(π‘₯,𝑦,𝑧,𝑔)

Proof of Theorem monfval
Dummy variables 𝑏 𝑐 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismon.s . 2 𝑀 = (Monoβ€˜πΆ)
2 ismon.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
3 fvexd 6854 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) ∈ V)
4 fveq2 6839 . . . . . 6 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
5 ismon.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
64, 5eqtr4di 2795 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
7 fvexd 6854 . . . . . 6 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) ∈ V)
8 simpl 483 . . . . . . . 8 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ 𝑐 = 𝐢)
98fveq2d 6843 . . . . . . 7 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
10 ismon.h . . . . . . 7 𝐻 = (Hom β€˜πΆ)
119, 10eqtr4di 2795 . . . . . 6 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) = 𝐻)
12 simplr 767 . . . . . . 7 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ 𝑏 = 𝐡)
13 simpr 485 . . . . . . . . 9 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ β„Ž = 𝐻)
1413oveqd 7368 . . . . . . . 8 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (π‘₯β„Žπ‘¦) = (π‘₯𝐻𝑦))
1513oveqd 7368 . . . . . . . . . . . 12 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (π‘§β„Žπ‘₯) = (𝑧𝐻π‘₯))
16 simpll 765 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ 𝑐 = 𝐢)
1716fveq2d 6843 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (compβ€˜π‘) = (compβ€˜πΆ))
18 ismon.o . . . . . . . . . . . . . . 15 Β· = (compβ€˜πΆ)
1917, 18eqtr4di 2795 . . . . . . . . . . . . . 14 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (compβ€˜π‘) = Β· )
2019oveqd 7368 . . . . . . . . . . . . 13 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦) = (βŸ¨π‘§, π‘₯⟩ Β· 𝑦))
2120oveqd 7368 . . . . . . . . . . . 12 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔) = (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))
2215, 21mpteq12dv 5194 . . . . . . . . . . 11 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔)))
2322cnveqd 5829 . . . . . . . . . 10 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔)) = β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔)))
2423funeqd 6520 . . . . . . . . 9 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔)) ↔ Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))))
2512, 24raleqbidv 3317 . . . . . . . 8 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔)) ↔ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))))
2614, 25rabeqbidv 3422 . . . . . . 7 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ {𝑓 ∈ (π‘₯β„Žπ‘¦) ∣ βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔))} = {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))})
2712, 12, 26mpoeq123dv 7426 . . . . . 6 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (π‘₯β„Žπ‘¦) ∣ βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔))}) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
287, 11, 27csbied2 3893 . . . . 5 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ ⦋(Hom β€˜π‘) / β„Žβ¦Œ(π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (π‘₯β„Žπ‘¦) ∣ βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔))}) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
293, 6, 28csbied2 3893 . . . 4 (𝑐 = 𝐢 β†’ ⦋(Baseβ€˜π‘) / π‘β¦Œβ¦‹(Hom β€˜π‘) / β„Žβ¦Œ(π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (π‘₯β„Žπ‘¦) ∣ βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔))}) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
30 df-mon 17573 . . . 4 Mono = (𝑐 ∈ Cat ↦ ⦋(Baseβ€˜π‘) / π‘β¦Œβ¦‹(Hom β€˜π‘) / β„Žβ¦Œ(π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (π‘₯β„Žπ‘¦) ∣ βˆ€π‘§ ∈ 𝑏 Fun β—‘(𝑔 ∈ (π‘§β„Žπ‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦)𝑔))}))
315fvexi 6853 . . . . 5 𝐡 ∈ V
3231, 31mpoex 8004 . . . 4 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}) ∈ V
3329, 30, 32fvmpt 6945 . . 3 (𝐢 ∈ Cat β†’ (Monoβ€˜πΆ) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
342, 33syl 17 . 2 (πœ‘ β†’ (Monoβ€˜πΆ) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
351, 34eqtrid 2789 1 (πœ‘ β†’ 𝑀 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ {𝑓 ∈ (π‘₯𝐻𝑦) ∣ βˆ€π‘§ ∈ 𝐡 Fun β—‘(𝑔 ∈ (𝑧𝐻π‘₯) ↦ (𝑓(βŸ¨π‘§, π‘₯⟩ Β· 𝑦)𝑔))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3062  {crab 3405  Vcvv 3443  β¦‹csb 3853  βŸ¨cop 4590   ↦ cmpt 5186  β—‘ccnv 5630  Fun wfun 6487  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  Basecbs 17043  Hom chom 17104  compcco 17105  Catccat 17504  Monocmon 17571
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-mon 17573
This theorem is referenced by:  ismon  17576  monpropd  17580
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