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Theorem bj-endval 36499
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 36498 . . 3 End = (𝑐 ∈ Cat ↦ (π‘₯ ∈ (Baseβ€˜π‘) ↦ {⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩, ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩}))
2 fveq2 6890 . . . 4 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
3 fveq2 6890 . . . . . . 7 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
43oveqd 7428 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘₯(Hom β€˜π‘)π‘₯) = (π‘₯(Hom β€˜πΆ)π‘₯))
54opeq2d 4879 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜π‘)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩)
6 fveq2 6890 . . . . . . 7 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
76oveqd 7428 . . . . . 6 (𝑐 = 𝐢 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯) = (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯))
87opeq2d 4879 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜π‘)π‘₯)⟩ = ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩)
95, 8preq12d 4744 . . . 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 5238 . . 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 6903 . . . . 5 (Baseβ€˜πΆ) ∈ V
1312mptex 7226 . . . 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 7020 . 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 7429 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯(Hom β€˜πΆ)π‘₯) = (𝑋(Hom β€˜πΆ)𝑋))
1817opeq2d 4879 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(Baseβ€˜ndx), (π‘₯(Hom β€˜πΆ)π‘₯)⟩ = ⟨(Baseβ€˜ndx), (𝑋(Hom β€˜πΆ)𝑋)⟩)
1916, 16opeq12d 4880 . . . . . 6 (π‘₯ = 𝑋 β†’ ⟨π‘₯, π‘₯⟩ = βŸ¨π‘‹, π‘‹βŸ©)
2019, 16oveq12d 7429 . . . . 5 (π‘₯ = 𝑋 β†’ (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯) = (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋))
2120opeq2d 4879 . . . 4 (π‘₯ = 𝑋 β†’ ⟨(+gβ€˜ndx), (⟨π‘₯, π‘₯⟩(compβ€˜πΆ)π‘₯)⟩ = ⟨(+gβ€˜ndx), (βŸ¨π‘‹, π‘‹βŸ©(compβ€˜πΆ)𝑋)⟩)
2218, 21preq12d 4744 . . 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 480 . 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 5431 . . 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 7004 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 1539   ∈ wcel 2104  Vcvv 3472  {cpr 4629  βŸ¨cop 4633   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411  ndxcnx 17130  Basecbs 17148  +gcplusg 17201  Hom chom 17212  compcco 17213  Catccat 17612  End cend 36497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-bj-end 36498
This theorem is referenced by:  bj-endbase  36500  bj-endcomp  36501
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