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Theorem catcfucclOLD 18094
Description: Obsolete proof of catcfuccl 18093 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 2727 . . . . 5 (𝑋 Func π‘Œ) = (𝑋 Func π‘Œ)
3 eqid 2727 . . . . 5 (𝑋 Nat π‘Œ) = (𝑋 Nat π‘Œ)
4 eqid 2727 . . . . 5 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
5 eqid 2727 . . . . 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 18075 . . . . . . 7 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Cat))
116, 10eleqtrd 2830 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (π‘ˆ ∩ Cat))
1211elin2d 4195 . . . . 5 (πœ‘ β†’ 𝑋 ∈ Cat)
13 catcfuccl.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝐡)
1413, 10eleqtrd 2830 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (π‘ˆ ∩ Cat))
1514elin2d 4195 . . . . 5 (πœ‘ β†’ π‘Œ ∈ Cat)
16 eqidd 2728 . . . . 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 17934 . . . 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 17166 . . . . . . 7 Base = Slot 1
19 catcfuccl.1 . . . . . . . 8 (πœ‘ β†’ Ο‰ ∈ π‘ˆ)
209, 19wunndx 17149 . . . . . . 7 (πœ‘ β†’ ndx ∈ π‘ˆ)
2118, 9, 20wunstr 17142 . . . . . 6 (πœ‘ β†’ (Baseβ€˜ndx) ∈ π‘ˆ)
2211elin1d 4194 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
2314elin1d 4194 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
249, 22, 23wunfunc 17872 . . . . . 6 (πœ‘ β†’ (𝑋 Func π‘Œ) ∈ π‘ˆ)
259, 21, 24wunop 10731 . . . . 5 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝑋 Func π‘Œ)⟩ ∈ π‘ˆ)
26 df-hom 17242 . . . . . . 7 Hom = Slot 14
2726, 9, 20wunstr 17142 . . . . . 6 (πœ‘ β†’ (Hom β€˜ndx) ∈ π‘ˆ)
289, 22, 23wunnat 17931 . . . . . 6 (πœ‘ β†’ (𝑋 Nat π‘Œ) ∈ π‘ˆ)
299, 27, 28wunop 10731 . . . . 5 (πœ‘ β†’ ⟨(Hom β€˜ndx), (𝑋 Nat π‘Œ)⟩ ∈ π‘ˆ)
30 df-cco 17243 . . . . . . 7 comp = Slot 15
3130, 9, 20wunstr 17142 . . . . . 6 (πœ‘ β†’ (compβ€˜ndx) ∈ π‘ˆ)
329, 24, 24wunxp 10733 . . . . . . . 8 (πœ‘ β†’ ((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
339, 32, 24wunxp 10733 . . . . . . 7 (πœ‘ β†’ (((𝑋 Func π‘Œ) Γ— (𝑋 Func π‘Œ)) Γ— (𝑋 Func π‘Œ)) ∈ π‘ˆ)
3430, 9, 23wunstr 17142 . . . . . . . . . . . . . 14 (πœ‘ β†’ (compβ€˜π‘Œ) ∈ π‘ˆ)
359, 34wunrn 10738 . . . . . . . . . . . . 13 (πœ‘ β†’ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
369, 35wununi 10715 . . . . . . . . . . . 12 (πœ‘ β†’ βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
379, 36wunrn 10738 . . . . . . . . . . 11 (πœ‘ β†’ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
389, 37wununi 10715 . . . . . . . . . 10 (πœ‘ β†’ βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
399, 38wunpw 10716 . . . . . . . . 9 (πœ‘ β†’ 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ π‘ˆ)
4018, 9, 22wunstr 17142 . . . . . . . . 9 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ π‘ˆ)
419, 39, 40wunmap 10735 . . . . . . . 8 (πœ‘ β†’ (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ π‘ˆ)
429, 28wunrn 10738 . . . . . . . . . 10 (πœ‘ β†’ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
439, 42wununi 10715 . . . . . . . . 9 (πœ‘ β†’ βˆͺ ran (𝑋 Nat π‘Œ) ∈ π‘ˆ)
449, 43, 43wunxp 10733 . . . . . . . 8 (πœ‘ β†’ (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ π‘ˆ)
459, 41, 44wunpm 10734 . . . . . . 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 6904 . . . . . . . . . . 11 (1st β€˜π‘£) ∈ V
47 fvex 6904 . . . . . . . . . . . . . 14 (2nd β€˜π‘£) ∈ V
48 ovex 7447 . . . . . . . . . . . . . . . . 17 (𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ V
49 ovex 7447 . . . . . . . . . . . . . . . . . . . 20 (𝑋 Nat π‘Œ) ∈ V
5049rnex 7910 . . . . . . . . . . . . . . . . . . 19 ran (𝑋 Nat π‘Œ) ∈ V
5150uniex 7738 . . . . . . . . . . . . . . . . . 18 βˆͺ ran (𝑋 Nat π‘Œ) ∈ V
5251, 51xpex 7747 . . . . . . . . . . . . . . . . 17 (βˆͺ ran (𝑋 Nat π‘Œ) Γ— βˆͺ ran (𝑋 Nat π‘Œ)) ∈ V
53 eqid 2727 . . . . . . . . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . . . . . . . . . . . . . . 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 5935 . . . . . . . . . . . . . . . . . . . . . . . . 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 4911 . . . . . . . . . . . . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . . . . . . . . . . . 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 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‘₯)) ∈ V
6160elpw 4602 . . . . . . . . . . . . . . . . . . . . . . 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 7116 . . . . . . . . . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (compβ€˜π‘Œ) ∈ V
6665rnex 7910 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran (compβ€˜π‘Œ) ∈ V
6766uniex 7738 . . . . . . . . . . . . . . . . . . . . . . . 24 βˆͺ ran (compβ€˜π‘Œ) ∈ V
6867rnex 7910 . . . . . . . . . . . . . . . . . . . . . . 23 ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
6968uniex 7738 . . . . . . . . . . . . . . . . . . . . . 22 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
7069pwex 5374 . . . . . . . . . . . . . . . . . . . . 21 𝒫 βˆͺ ran βˆͺ ran (compβ€˜π‘Œ) ∈ V
71 fvex 6904 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜π‘‹) ∈ V
7270, 71elmap 8879 . . . . . . . . . . . . . . . . . . . 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 3061 . . . . . . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . . . . . . . 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 8064 . . . . . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . . . . . . . 18 (𝑔(𝑋 Nat π‘Œ)β„Ž) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
79 ovssunirn 7450 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑋 Nat π‘Œ)𝑔) βŠ† βˆͺ ran (𝑋 Nat π‘Œ)
80 xpss12 5687 . . . . . . . . . . . . . . . . . 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 8853 . . . . . . . . . . . . . . . . 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 3789 . . . . . . . . . . . . . . 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 4409 . . . . . . . . . . . . . . 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 3789 . . . . . . . . . . . 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 4409 . . . . . . . . . . . 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 3061 . . . . . . . . 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 2727 . . . . . . . . . 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 8064 . . . . . . . . 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 10736 . . . . . 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 10731 . . . . 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 10720 . . . 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 2828 . . 3 (πœ‘ β†’ 𝑄 ∈ π‘ˆ)
1011, 12, 15fuccat 17947 . . 3 (πœ‘ β†’ 𝑄 ∈ Cat)
102100, 101elind 4190 . 2 (πœ‘ β†’ 𝑄 ∈ (π‘ˆ ∩ Cat))
103102, 10eleqtrrd 2831 1 (πœ‘ β†’ 𝑄 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  Vcvv 3469  [wsbc 3774  β¦‹csb 3889   ∩ cin 3943   βŠ† wss 3944  π’« cpw 4598  {ctp 4628  βŸ¨cop 4630  βˆͺ cuni 4903   ↦ cmpt 5225   Γ— cxp 5670  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Ο‰com 7862  1st c1st 7983  2nd c2nd 7984   ↑m cmap 8834   ↑pm cpm 8835  WUnicwun 10709  1c1 11125  4c4 12285  5c5 12286  cdc 12693  ndxcnx 17147  Basecbs 17165  Hom chom 17229  compcco 17230  Catccat 17629   Func cfunc 17825   Nat cnat 17916   FuncCat cfuc 17917  CatCatccatc 18072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-omul 8483  df-er 8716  df-ec 8718  df-qs 8722  df-map 8836  df-pm 8837  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-wun 10711  df-ni 10881  df-pli 10882  df-mi 10883  df-lti 10884  df-plpq 10917  df-mpq 10918  df-ltpq 10919  df-enq 10920  df-nq 10921  df-erq 10922  df-plq 10923  df-mq 10924  df-1nq 10925  df-rq 10926  df-ltnq 10927  df-np 10990  df-plp 10992  df-ltp 10994  df-enr 11064  df-nr 11065  df-c 11130  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-slot 17136  df-ndx 17148  df-base 17166  df-hom 17242  df-cco 17243  df-cat 17633  df-cid 17634  df-func 17829  df-nat 17918  df-fuc 17919  df-catc 18073
This theorem is referenced by: (None)
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