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Theorem bj-endval 36196
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 36195 . . 3 End = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
2 fveq2 6892 . . . 4 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
3 fveq2 6892 . . . . . . 7 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
43oveqd 7426 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘₯(Hom β€˜π‘)π‘₯) = (π‘₯(Hom β€˜πΆ)π‘₯))
54opeq2d 4881 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩)
6 fveq2 6892 . . . . . . 7 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
76oveqd 7426 . . . . . 6 (𝑐 = 𝐢 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯) = (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯))
87opeq2d 4881 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩ = ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩)
95, 8preq12d 4746 . . . 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 5240 . . 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 6905 . . . . 5 (Baseβ€˜πΆ) ∈ V
1312mptex 7225 . . . 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 7022 . 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 7427 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯(Hom β€˜πΆ)π‘₯) = (𝑋(Hom β€˜πΆ)𝑋))
1817opeq2d 4881 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩)
1916, 16opeq12d 4882 . . . . . 6 (π‘₯ = 𝑋 β†’ ⟨π‘₯, π‘₯⟩ = βŸ¨π‘‹, π‘‹βŸ©)
2019, 16oveq12d 7427 . . . . 5 (π‘₯ = 𝑋 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯) = (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋))
2120opeq2d 4881 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩ = ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩)
2218, 21preq12d 4746 . . 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 483 . 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 5433 . . 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 7006 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 1542   ∈ wcel 2107  Vcvv 3475  {cpr 4631  βŸ¨cop 4635   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  Hom chom 17208  compcco 17209  Catccat 17608  End cend 36194
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-pr 5428
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-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-bj-end 36195
This theorem is referenced by:  bj-endbase  36197  bj-endcomp  36198
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