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Theorem bj-endval 36500
Description: Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-endval.c (πœ‘ β†’ 𝐢 ∈ Cat)
bj-endval.x (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜πΆ))
Assertion
Ref Expression
bj-endval (πœ‘ β†’ ((End β€˜πΆ)β€˜π‘‹) = {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩})

Proof of Theorem bj-endval
Dummy variables 𝑐 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-bj-end 36499 . . 3 End = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
2 fveq2 6891 . . . 4 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
3 fveq2 6891 . . . . . . 7 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
43oveqd 7429 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘₯(Hom β€˜π‘)π‘₯) = (π‘₯(Hom β€˜πΆ)π‘₯))
54opeq2d 4880 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩)
6 fveq2 6891 . . . . . . 7 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
76oveqd 7429 . . . . . 6 (𝑐 = 𝐢 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯) = (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯))
87opeq2d 4880 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩ = ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩)
95, 8preq12d 4745 . . . 4 (𝑐 = 𝐢 β†’ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩} = {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩})
102, 9mpteq12dv 5239 . . 3 (𝑐 = 𝐢 β†’ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}) = (π‘₯ ∈ (Baseβ€˜πΆ) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩}))
11 bj-endval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
12 fvex 6904 . . . . 5 (Baseβ€˜πΆ) ∈ V
1312mptex 7227 . . . 4 (π‘₯ ∈ (Baseβ€˜πΆ) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩}) ∈ V
1413a1i 11 . . 3 (πœ‘ β†’ (π‘₯ ∈ (Baseβ€˜πΆ) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩}) ∈ V)
151, 10, 11, 14fvmptd3 7021 . 2 (πœ‘ β†’ (End β€˜πΆ) = (π‘₯ ∈ (Baseβ€˜πΆ) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩}))
16 id 22 . . . . . 6 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
1716, 16oveq12d 7430 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯(Hom β€˜πΆ)π‘₯) = (𝑋(Hom β€˜πΆ)𝑋))
1817opeq2d 4880 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩)
1916, 16opeq12d 4881 . . . . . 6 (π‘₯ = 𝑋 β†’ ⟨π‘₯, π‘₯⟩ = βŸ¨π‘‹, π‘‹βŸ©)
2019, 16oveq12d 7430 . . . . 5 (π‘₯ = 𝑋 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯) = (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋))
2120opeq2d 4880 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩ = ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩)
2218, 21preq12d 4745 . . 3 (π‘₯ = 𝑋 β†’ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩} = {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩})
2322adantl 481 . 2 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩} = {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩})
24 bj-endval.x . 2 (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜πΆ))
25 prex 5432 . . 3 {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩} ∈ V
2625a1i 11 . 2 (πœ‘ β†’ {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩} ∈ V)
2715, 23, 24, 26fvmptd 7005 1 (πœ‘ β†’ ((End β€˜πΆ)β€˜π‘‹) = {⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩, ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473  {cpr 4630  βŸ¨cop 4634   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7412  ndxcnx 17131  Basecbs 17149  +gcplusg 17202  Hom chom 17213  compcco 17214  Catccat 17613  End cend 36498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-bj-end 36499
This theorem is referenced by:  bj-endbase  36501  bj-endcomp  36502
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