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Theorem catcfucclOLD 18011
Description: Obsolete proof of catcfuccl 18010 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
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
catcfucclOLD (πœ‘ β†’ 𝑄 ∈ 𝐡)

Proof of Theorem catcfucclOLD
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 2733 . . . . 5 (𝑋 Func π‘Œ) = (𝑋 Func π‘Œ)
3 eqid 2733 . . . . 5 (𝑋 Nat π‘Œ) = (𝑋 Nat π‘Œ)
4 eqid 2733 . . . . 5 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
5 eqid 2733 . . . . 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 17992 . . . . . . 7 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
116, 10eleqtrd 2836 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1211elin2d 4160 . . . . 5 (πœ‘ β†’ 𝑋 ∈ Cat)
13 catcfuccl.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝐡)
1413, 10eleqtrd 2836 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (π‘ˆ ∩ Cat))
1514elin2d 4160 . . . . 5 (πœ‘ β†’ π‘Œ ∈ Cat)
16 eqidd 2734 . . . . 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 17851 . . . 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 df-base 17089 . . . . . . 7 Base = Slot 1
19 catcfuccl.1 . . . . . . . 8 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
209, 19wunndx 17072 . . . . . . 7 (πœ‘ β†’ ndx ∈ π‘ˆ)
2118, 9, 20wunstr 17065 . . . . . 6 (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
2211elin1d 4159 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
2314elin1d 4159 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
249, 22, 23wunfunc 17790 . . . . . 6 (πœ‘ β†’ (𝑋 Func π‘Œ) ∈ π‘ˆ)
259, 21, 24wunop 10663 . . . . 5 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝑋 Func π‘Œ)⟩ ∈ π‘ˆ)
26 df-hom 17162 . . . . . . 7 Hom = Slot 14
2726, 9, 20wunstr 17065 . . . . . 6 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
289, 22, 23wunnat 17848 . . . . . 6 (πœ‘ β†’ (𝑋 Nat π‘Œ) ∈ π‘ˆ)
299, 27, 28wunop 10663 . . . . 5 (πœ‘ β†’ ⟨(Hom β€˜ndx), (𝑋 Nat π‘Œ)⟩ ∈ π‘ˆ)
30 df-cco 17163 . . . . . . 7 comp = Slot 15
3130, 9, 20wunstr 17065 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
329, 24, 24wunxp 10665 . . . . . . . 8 (πœ‘ β†’ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
339, 32, 24wunxp 10665 . . . . . . 7 (πœ‘ β†’ (((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
3430, 9, 23wunstr 17065 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘Œ) ∈ π‘ˆ)
359, 34wunrn 10670 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
369, 35wununi 10647 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
379, 36wunrn 10670 . . . . . . . . . . 11 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
389, 37wununi 10647 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
399, 38wunpw 10648 . . . . . . . . 9 (πœ‘ β†’ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
4018, 9, 22wunstr 17065 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
419, 39, 40wunmap 10667 . . . . . . . 8 (πœ‘ β†’ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ π‘ˆ)
429, 28wunrn 10670 . . . . . . . . . 10 (πœ‘ β†’ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
439, 42wununi 10647 . . . . . . . . 9 (πœ‘ β†’ βˆͺ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
449, 43, 43wunxp 10665 . . . . . . . 8 (πœ‘ β†’ (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ π‘ˆ)
459, 41, 44wunpm 10666 . . . . . . 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 6856 . . . . . . . . . . 11 (1st β€˜π‘£) ∈ V
47 fvex 6856 . . . . . . . . . . . . . 14 (2nd β€˜π‘£) ∈ V
48 ovex 7391 . . . . . . . . . . . . . . . . 17 (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ V
49 ovex 7391 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat π‘Œ) ∈ V
5049rnex 7850 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat π‘Œ) ∈ V
5150uniex 7679 . . . . . . . . . . . . . . . . . 18 βˆͺ ran (𝑋 Nat π‘Œ) ∈ V
5251, 51xpex 7688 . . . . . . . . . . . . . . . . 17 (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ V
53 eqid 2733 . . . . . . . . . . . . . . . . . . . . 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 7394 . . . . . . . . . . . . . . . . . . . . . . . 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 7394 . . . . . . . . . . . . . . . . . . . . . . . . 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 5895 . . . . . . . . . . . . . . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . . . . . . . . . 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 7391 . . . . . . . . . . . . . . . . . . . . . . . 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 4565 . . . . . . . . . . . . . . . . . . . . . . 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 7061 . . . . . . . . . . . . . . . . . . . 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 6856 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (compβ€˜π‘Œ) ∈ V
6665rnex 7850 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (compβ€˜π‘Œ) ∈ V
6766uniex 7679 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆͺ ran (compβ€˜π‘Œ) ∈ V
6867rnex 7850 . . . . . . . . . . . . . . . . . . . . . . 23 ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
6968uniex 7679 . . . . . . . . . . . . . . . . . . . . . 22 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
7069pwex 5336 . . . . . . . . . . . . . . . . . . . . 21 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
71 fvex 6856 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜π‘‹) ∈ V
7270, 71elmap 8812 . . . . . . . . . . . . . . . . . . . 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 3066 . . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . . 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 8001 . . . . . . . . . . . . . . . . . 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 7394 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat π‘Œ)β„Ž) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
79 ovssunirn 7394 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat π‘Œ)𝑔) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
80 xpss12 5649 . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . 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 8786 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 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 3755 . . . . . . . . . . . . . . 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 4374 . . . . . . . . . . . . . . 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 3755 . . . . . . . . . . . 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 4374 . . . . . . . . . . . 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 3066 . . . . . . . . 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 2733 . . . . . . . . . 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 8001 . . . . . . . . 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 10668 . . . . . 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 10663 . . . . 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 10652 . . . 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 2834 . . 3 (πœ‘ β†’ 𝑄 ∈ π‘ˆ)
1011, 12, 15fuccat 17864 . . 3 (πœ‘ β†’ 𝑄 ∈ Cat)
102100, 101elind 4155 . 2 (πœ‘ β†’ 𝑄 ∈ (π‘ˆ ∩ Cat))
103102, 10eleqtrrd 2837 1 (πœ‘ β†’ 𝑄 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444  [wsbc 3740  β¦‹csb 3856   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  {ctp 4591  βŸ¨cop 4593  βˆͺ cuni 4866   ↦ cmpt 5189   Γ— cxp 5632  ran crn 5635  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  Ο‰com 7803  1st c1st 7920  2nd c2nd 7921   ↑m cmap 8768   ↑pm cpm 8769  WUnicwun 10641  1c1 11057  4c4 12215  5c5 12216  cdc 12623  ndxcnx 17070  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549   Func cfunc 17745   Nat cnat 17833   FuncCat cfuc 17834  CatCatccatc 17989
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-omul 8418  df-er 8651  df-ec 8653  df-qs 8657  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-wun 10643  df-ni 10813  df-pli 10814  df-mi 10815  df-lti 10816  df-plpq 10849  df-mpq 10850  df-ltpq 10851  df-enq 10852  df-nq 10853  df-erq 10854  df-plq 10855  df-mq 10856  df-1nq 10857  df-rq 10858  df-ltnq 10859  df-np 10922  df-plp 10924  df-ltp 10926  df-enr 10996  df-nr 10997  df-c 11062  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-nat 17835  df-fuc 17836  df-catc 17990
This theorem is referenced by: (None)
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