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Theorem monfval 17679
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 6907 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) ∈ V)
4 fveq2 6892 . . . . . 6 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
5 ismon.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
64, 5eqtr4di 2791 . . . . 5 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
7 fvexd 6907 . . . . . 6 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) ∈ V)
8 simpl 484 . . . . . . . 8 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ 𝑐 = 𝐢)
98fveq2d 6896 . . . . . . 7 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
10 ismon.h . . . . . . 7 𝐻 = (Hom β€˜πΆ)
119, 10eqtr4di 2791 . . . . . 6 ((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) β†’ (Hom β€˜π‘) = 𝐻)
12 simplr 768 . . . . . . 7 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ 𝑏 = 𝐡)
13 simpr 486 . . . . . . . . 9 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ β„Ž = 𝐻)
1413oveqd 7426 . . . . . . . 8 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (π‘₯β„Žπ‘¦) = (π‘₯𝐻𝑦))
1513oveqd 7426 . . . . . . . . . . . 12 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (π‘§β„Žπ‘₯) = (𝑧𝐻π‘₯))
16 simpll 766 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ 𝑐 = 𝐢)
1716fveq2d 6896 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (compβ€˜π‘) = (compβ€˜πΆ))
18 ismon.o . . . . . . . . . . . . . . 15 Β· = (compβ€˜πΆ)
1917, 18eqtr4di 2791 . . . . . . . . . . . . . 14 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (compβ€˜π‘) = Β· )
2019oveqd 7426 . . . . . . . . . . . . 13 (((𝑐 = 𝐢 ∧ 𝑏 = 𝐡) ∧ β„Ž = 𝐻) β†’ (βŸ¨π‘§, π‘₯⟩(compβ€˜π‘)𝑦) = (βŸ¨π‘§, π‘₯⟩ Β· 𝑦))
2120oveqd 7426 . . . . . . . . . . . 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 5240 . . . . . . . . . . 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 5876 . . . . . . . . . 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 6571 . . . . . . . . 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 3343 . . . . . . . 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 3450 . . . . . . 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 7484 . . . . . 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 3934 . . . . 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 3934 . . . 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 17677 . . . 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 6906 . . . . 5 𝐡 ∈ V
3231, 31mpoex 8066 . . . 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 6999 . . 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 2785 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475  β¦‹csb 3894  βŸ¨cop 4635   ↦ cmpt 5232  β—‘ccnv 5676  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Monocmon 17675
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-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-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-mon 17677
This theorem is referenced by:  ismon  17680  monpropd  17684
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