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Theorem catcfuccl 18073
Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
Hypotheses
Ref Expression
catcfuccl.c 𝐢 = (CatCatβ€˜π‘ˆ)
catcfuccl.b 𝐡 = (Baseβ€˜πΆ)
catcfuccl.o 𝑄 = (𝑋 FuncCat π‘Œ)
catcfuccl.u (πœ‘ β†’ π‘ˆ ∈ WUni)
catcfuccl.1 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
catcfuccl.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
catcfuccl.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
catcfuccl (πœ‘ β†’ 𝑄 ∈ 𝐡)

Proof of Theorem catcfuccl
Dummy variables π‘Ž 𝑏 𝑓 𝑔 β„Ž 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcfuccl.o . . . . 5 𝑄 = (𝑋 FuncCat π‘Œ)
2 eqid 2730 . . . . 5 (𝑋 Func π‘Œ) = (𝑋 Func π‘Œ)
3 eqid 2730 . . . . 5 (𝑋 Nat π‘Œ) = (𝑋 Nat π‘Œ)
4 eqid 2730 . . . . 5 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
5 eqid 2730 . . . . 5 (compβ€˜π‘Œ) = (compβ€˜π‘Œ)
6 catcfuccl.x . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝐡)
7 catcfuccl.c . . . . . . . 8 𝐢 = (CatCatβ€˜π‘ˆ)
8 catcfuccl.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
9 catcfuccl.u . . . . . . . 8 (πœ‘ β†’ π‘ˆ ∈ WUni)
107, 8, 9catcbas 18055 . . . . . . 7 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
116, 10eleqtrd 2833 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1211elin2d 4198 . . . . 5 (πœ‘ β†’ 𝑋 ∈ Cat)
13 catcfuccl.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝐡)
1413, 10eleqtrd 2833 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (π‘ˆ ∩ Cat))
1514elin2d 4198 . . . . 5 (πœ‘ β†’ π‘Œ ∈ Cat)
16 eqidd 2731 . . . . 5 (πœ‘ β†’ (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))) = (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
171, 2, 3, 4, 5, 12, 15, 16fucval 17914 . . . 4 (πœ‘ β†’ 𝑄 = {⟨(Baseβ€˜ndx), (𝑋 Func π‘Œ)⟩, ⟨(Hom β€˜ndx), (𝑋 Nat π‘Œ)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩})
18 baseid 17151 . . . . . . 7 Base = Slot (Baseβ€˜ndx)
19 catcfuccl.1 . . . . . . . 8 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
209, 19wunndx 17132 . . . . . . 7 (πœ‘ β†’ ndx ∈ π‘ˆ)
2118, 9, 20wunstr 17125 . . . . . 6 (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
227, 8, 9, 6catcbascl 18066 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
237, 8, 9, 13catcbascl 18066 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
249, 22, 23wunfunc 17853 . . . . . 6 (πœ‘ β†’ (𝑋 Func π‘Œ) ∈ π‘ˆ)
259, 21, 24wunop 10719 . . . . 5 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝑋 Func π‘Œ)⟩ ∈ π‘ˆ)
26 homid 17361 . . . . . . 7 Hom = Slot (Hom β€˜ndx)
2726, 9, 20wunstr 17125 . . . . . 6 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
289, 22, 23wunnat 17911 . . . . . 6 (πœ‘ β†’ (𝑋 Nat π‘Œ) ∈ π‘ˆ)
299, 27, 28wunop 10719 . . . . 5 (πœ‘ β†’ ⟨(Hom β€˜ndx), (𝑋 Nat π‘Œ)⟩ ∈ π‘ˆ)
30 ccoid 17363 . . . . . . 7 comp = Slot (compβ€˜ndx)
3130, 9, 20wunstr 17125 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
329, 24, 24wunxp 10721 . . . . . . . 8 (πœ‘ β†’ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
339, 32, 24wunxp 10721 . . . . . . 7 (πœ‘ β†’ (((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
347, 8, 9, 13catcccocl 18070 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘Œ) ∈ π‘ˆ)
359, 34wunrn 10726 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
369, 35wununi 10703 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
379, 36wunrn 10726 . . . . . . . . . . 11 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
389, 37wununi 10703 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
399, 38wunpw 10704 . . . . . . . . 9 (πœ‘ β†’ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
407, 8, 9, 6catcbaselcl 18068 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
419, 39, 40wunmap 10723 . . . . . . . 8 (πœ‘ β†’ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ π‘ˆ)
429, 28wunrn 10726 . . . . . . . . . 10 (πœ‘ β†’ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
439, 42wununi 10703 . . . . . . . . 9 (πœ‘ β†’ βˆͺ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
449, 43, 43wunxp 10721 . . . . . . . 8 (πœ‘ β†’ (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ π‘ˆ)
459, 41, 44wunpm 10722 . . . . . . 7 (πœ‘ β†’ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))) ∈ π‘ˆ)
46 fvex 6903 . . . . . . . . . . 11 (1st β€˜π‘£) ∈ V
47 fvex 6903 . . . . . . . . . . . . . 14 (2nd β€˜π‘£) ∈ V
48 ovex 7444 . . . . . . . . . . . . . . . . 17 (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ V
49 ovex 7444 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat π‘Œ) ∈ V
5049rnex 7905 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat π‘Œ) ∈ V
5150uniex 7733 . . . . . . . . . . . . . . . . . 18 βˆͺ ran (𝑋 Nat π‘Œ) ∈ V
5251, 51xpex 7742 . . . . . . . . . . . . . . . . 17 (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ V
53 eqid 2730 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) = (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))
54 ovssunirn 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) βŠ† βˆͺ ran (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))
55 ovssunirn 7447 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘Œ)
56 rnss 5937 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† βˆͺ ran (compβ€˜π‘Œ) β†’ ran (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘Œ))
57 uniss 4915 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ran (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† ran βˆͺ ran (compβ€˜π‘Œ) β†’ βˆͺ ran (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran (compβ€˜π‘Œ))
5855, 56, 57mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆͺ ran (⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran (compβ€˜π‘Œ)
5954, 58sstri 3990 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran (compβ€˜π‘Œ)
60 ovex 7444 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) ∈ V
6160elpw 4605 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↔ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) βŠ† βˆͺ ran βˆͺ ran (compβ€˜π‘Œ))
6259, 61mpbir 230 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ)
6362a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ (Baseβ€˜π‘‹) β†’ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)) ∈ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ))
6453, 63fmpti 7112 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))):(Baseβ€˜π‘‹)βŸΆπ’« βˆͺ ran βˆͺ ran (compβ€˜π‘Œ)
65 fvex 6903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (compβ€˜π‘Œ) ∈ V
6665rnex 7905 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (compβ€˜π‘Œ) ∈ V
6766uniex 7733 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆͺ ran (compβ€˜π‘Œ) ∈ V
6867rnex 7905 . . . . . . . . . . . . . . . . . . . . . . 23 ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
6968uniex 7733 . . . . . . . . . . . . . . . . . . . . . 22 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
7069pwex 5377 . . . . . . . . . . . . . . . . . . . . 21 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
71 fvex 6903 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜π‘‹) ∈ V
7270, 71elmap 8867 . . . . . . . . . . . . . . . . . . . 20 ((π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↔ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))):(Baseβ€˜π‘‹)βŸΆπ’« βˆͺ ran βˆͺ ran (compβ€˜π‘Œ))
7364, 72mpbir 230 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))
7473rgen2w 3064 . . . . . . . . . . . . . . . . . 18 βˆ€π‘ ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž)βˆ€π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔)(π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))
75 eqid 2730 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) = (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))
7675fmpo 8056 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘ ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž)βˆ€π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔)(π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))) ∈ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↔ (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))):((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔))⟢(𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
7774, 76mpbi 229 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))):((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔))⟢(𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))
78 ovssunirn 7447 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat π‘Œ)β„Ž) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
79 ovssunirn 7447 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat π‘Œ)𝑔) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
80 xpss12 5690 . . . . . . . . . . . . . . . . . 18 (((𝑔(𝑋 Nat π‘Œ)β„Ž) βŠ† βˆͺ ran (𝑋 Nat π‘Œ) ∧ (𝑓(𝑋 Nat π‘Œ)𝑔) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)) β†’ ((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔)) βŠ† (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
8178, 79, 80mp2an 688 . . . . . . . . . . . . . . . . 17 ((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔)) βŠ† (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))
82 elpm2r 8841 . . . . . . . . . . . . . . . . 17 ((((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ V ∧ (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ V) ∧ ((𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))):((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔))⟢(𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∧ ((𝑔(𝑋 Nat π‘Œ)β„Ž) Γ— (𝑓(𝑋 Nat π‘Œ)𝑔)) βŠ† (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))) β†’ (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
8348, 52, 77, 81, 82mp4an 689 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
8483sbcth 3791 . . . . . . . . . . . . . . 15 ((2nd β€˜π‘£) ∈ V β†’ [(2nd β€˜π‘£) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
85 sbcel1g 4412 . . . . . . . . . . . . . . 15 ((2nd β€˜π‘£) ∈ V β†’ ([(2nd β€˜π‘£) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))) ↔ ⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))))
8684, 85mpbid 231 . . . . . . . . . . . . . 14 ((2nd β€˜π‘£) ∈ V β†’ ⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
8747, 86ax-mp 5 . . . . . . . . . . . . 13 ⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
8887sbcth 3791 . . . . . . . . . . . 12 ((1st β€˜π‘£) ∈ V β†’ [(1st β€˜π‘£) / 𝑓]⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
89 sbcel1g 4412 . . . . . . . . . . . 12 ((1st β€˜π‘£) ∈ V β†’ ([(1st β€˜π‘£) / 𝑓]⦋(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))) ↔ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))))
9088, 89mpbid 231 . . . . . . . . . . 11 ((1st β€˜π‘£) ∈ V β†’ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
9146, 90ax-mp 5 . . . . . . . . . 10 ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
9291rgen2w 3064 . . . . . . . . 9 βˆ€π‘£ ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ))βˆ€β„Ž ∈ (𝑋 Func π‘Œ)⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
93 eqid 2730 . . . . . . . . . 10 (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))) = (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))
9493fmpo 8056 . . . . . . . . 9 (βˆ€π‘£ ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ))βˆ€β„Ž ∈ (𝑋 Func π‘Œ)⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))) ∈ ((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))) ↔ (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))):(((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ))⟢((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
9592, 94mpbi 229 . . . . . . . 8 (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))):(((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ))⟢((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)))
9695a1i 11 . . . . . . 7 (πœ‘ β†’ (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))):(((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ))⟢((𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↑pm (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ))))
979, 33, 45, 96wunf 10724 . . . . . 6 (πœ‘ β†’ (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))) ∈ π‘ˆ)
989, 31, 97wunop 10719 . . . . 5 (πœ‘ β†’ ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩ ∈ π‘ˆ)
999, 25, 29, 98wuntp 10708 . . . 4 (πœ‘ β†’ {⟨(Baseβ€˜ndx), (𝑋 Func π‘Œ)⟩, ⟨(Hom β€˜ndx), (𝑋 Nat π‘Œ)⟩, ⟨(compβ€˜ndx), (𝑣 ∈ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)), β„Ž ∈ (𝑋 Func π‘Œ) ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (𝑔(𝑋 Nat π‘Œ)β„Ž), π‘Ž ∈ (𝑓(𝑋 Nat π‘Œ)𝑔) ↦ (π‘₯ ∈ (Baseβ€˜π‘‹) ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩(compβ€˜π‘Œ)((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯)))))⟩} ∈ π‘ˆ)
10017, 99eqeltrd 2831 . . 3 (πœ‘ β†’ 𝑄 ∈ π‘ˆ)
1011, 12, 15fuccat 17927 . . 3 (πœ‘ β†’ 𝑄 ∈ Cat)
102100, 101elind 4193 . 2 (πœ‘ β†’ 𝑄 ∈ (π‘ˆ ∩ Cat))
103102, 10eleqtrrd 2834 1 (πœ‘ β†’ 𝑄 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472  [wsbc 3776  β¦‹csb 3892   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  {ctp 4631  βŸ¨cop 4633  βˆͺ cuni 4907   ↦ cmpt 5230   Γ— cxp 5673  ran crn 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  Ο‰com 7857  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822   ↑pm cpm 8823  WUnicwun 10697  ndxcnx 17130  Basecbs 17148  Hom chom 17212  compcco 17213  Catccat 17612   Func cfunc 17808   Nat cnat 17896   FuncCat cfuc 17897  CatCatccatc 18052
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-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  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-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  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-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  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-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-wun 10699  df-ni 10869  df-pli 10870  df-mi 10871  df-lti 10872  df-plpq 10905  df-mpq 10906  df-ltpq 10907  df-enq 10908  df-nq 10909  df-erq 10910  df-plq 10911  df-mq 10912  df-1nq 10913  df-rq 10914  df-ltnq 10915  df-np 10978  df-plp 10980  df-ltp 10982  df-enr 11052  df-nr 11053  df-c 11118  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-hom 17225  df-cco 17226  df-cat 17616  df-cid 17617  df-func 17812  df-nat 17898  df-fuc 17899  df-catc 18053
This theorem is referenced by: (None)
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