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Theorem wunnat 17904
Description: A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
Hypotheses
Ref Expression
wunnat.1 (πœ‘ β†’ π‘ˆ ∈ WUni)
wunnat.2 (πœ‘ β†’ 𝐢 ∈ π‘ˆ)
wunnat.3 (πœ‘ β†’ 𝐷 ∈ π‘ˆ)
Assertion
Ref Expression
wunnat (πœ‘ β†’ (𝐢 Nat 𝐷) ∈ π‘ˆ)

Proof of Theorem wunnat
Dummy variables 𝑓 π‘Ž 𝑔 π‘Ÿ 𝑠 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunnat.1 . 2 (πœ‘ β†’ π‘ˆ ∈ WUni)
2 wunnat.2 . . . 4 (πœ‘ β†’ 𝐢 ∈ π‘ˆ)
3 wunnat.3 . . . 4 (πœ‘ β†’ 𝐷 ∈ π‘ˆ)
41, 2, 3wunfunc 17846 . . 3 (πœ‘ β†’ (𝐢 Func 𝐷) ∈ π‘ˆ)
51, 4, 4wunxp 10716 . 2 (πœ‘ β†’ ((𝐢 Func 𝐷) Γ— (𝐢 Func 𝐷)) ∈ π‘ˆ)
6 homid 17354 . . . . . . 7 Hom = Slot (Hom β€˜ndx)
76, 1, 3wunstr 17118 . . . . . 6 (πœ‘ β†’ (Hom β€˜π·) ∈ π‘ˆ)
81, 7wunrn 10721 . . . . 5 (πœ‘ β†’ ran (Hom β€˜π·) ∈ π‘ˆ)
91, 8wununi 10698 . . . 4 (πœ‘ β†’ βˆͺ ran (Hom β€˜π·) ∈ π‘ˆ)
10 baseid 17144 . . . . 5 Base = Slot (Baseβ€˜ndx)
1110, 1, 2wunstr 17118 . . . 4 (πœ‘ β†’ (Baseβ€˜πΆ) ∈ π‘ˆ)
121, 9, 11wunmap 10718 . . 3 (πœ‘ β†’ (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ∈ π‘ˆ)
131, 12wunpw 10699 . 2 (πœ‘ β†’ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ∈ π‘ˆ)
14 fvex 6902 . . . . . 6 (1st β€˜π‘“) ∈ V
15 fvex 6902 . . . . . . . . 9 (1st β€˜π‘”) ∈ V
16 ovex 7439 . . . . . . . . . . . 12 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ∈ V
17 ssrab2 4077 . . . . . . . . . . . . 13 {π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} βŠ† Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯))
18 ovssunirn 7442 . . . . . . . . . . . . . . . 16 ((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† βˆͺ ran (Hom β€˜π·)
1918rgenw 3066 . . . . . . . . . . . . . . 15 βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† βˆͺ ran (Hom β€˜π·)
20 ss2ixp 8901 . . . . . . . . . . . . . . 15 (βˆ€π‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† βˆͺ ran (Hom β€˜π·) β†’ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† Xπ‘₯ ∈ (Baseβ€˜πΆ)βˆͺ ran (Hom β€˜π·))
2119, 20ax-mp 5 . . . . . . . . . . . . . 14 Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† Xπ‘₯ ∈ (Baseβ€˜πΆ)βˆͺ ran (Hom β€˜π·)
22 fvex 6902 . . . . . . . . . . . . . . 15 (Baseβ€˜πΆ) ∈ V
23 fvex 6902 . . . . . . . . . . . . . . . . 17 (Hom β€˜π·) ∈ V
2423rnex 7900 . . . . . . . . . . . . . . . 16 ran (Hom β€˜π·) ∈ V
2524uniex 7728 . . . . . . . . . . . . . . 15 βˆͺ ran (Hom β€˜π·) ∈ V
2622, 25ixpconst 8898 . . . . . . . . . . . . . 14 Xπ‘₯ ∈ (Baseβ€˜πΆ)βˆͺ ran (Hom β€˜π·) = (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
2721, 26sseqtri 4018 . . . . . . . . . . . . 13 Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) βŠ† (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
2817, 27sstri 3991 . . . . . . . . . . . 12 {π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} βŠ† (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
2916, 28elpwi2 5346 . . . . . . . . . . 11 {π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
3029sbcth 3792 . . . . . . . . . 10 ((1st β€˜π‘”) ∈ V β†’ [(1st β€˜π‘”) / 𝑠]{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
31 sbcel1g 4413 . . . . . . . . . 10 ((1st β€˜π‘”) ∈ V β†’ ([(1st β€˜π‘”) / 𝑠]{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ↔ ⦋(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))))
3230, 31mpbid 231 . . . . . . . . 9 ((1st β€˜π‘”) ∈ V β†’ ⦋(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
3315, 32ax-mp 5 . . . . . . . 8 ⦋(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
3433sbcth 3792 . . . . . . 7 ((1st β€˜π‘“) ∈ V β†’ [(1st β€˜π‘“) / π‘Ÿ]⦋(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
35 sbcel1g 4413 . . . . . . 7 ((1st β€˜π‘“) ∈ V β†’ ([(1st β€˜π‘“) / π‘Ÿ]⦋(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ↔ ⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))))
3634, 35mpbid 231 . . . . . 6 ((1st β€˜π‘“) ∈ V β†’ ⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
3714, 36ax-mp 5 . . . . 5 ⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
3837rgen2w 3067 . . . 4 βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘” ∈ (𝐢 Func 𝐷)⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
39 eqid 2733 . . . . . 6 (𝐢 Nat 𝐷) = (𝐢 Nat 𝐷)
40 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
41 eqid 2733 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
42 eqid 2733 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
43 eqid 2733 . . . . . 6 (compβ€˜π·) = (compβ€˜π·)
4439, 40, 41, 42, 43natfval 17894 . . . . 5 (𝐢 Nat 𝐷) = (𝑓 ∈ (𝐢 Func 𝐷), 𝑔 ∈ (𝐢 Func 𝐷) ↦ ⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))})
4544fmpo 8051 . . . 4 (βˆ€π‘“ ∈ (𝐢 Func 𝐷)βˆ€π‘” ∈ (𝐢 Func 𝐷)⦋(1st β€˜π‘“) / π‘Ÿβ¦Œβ¦‹(1st β€˜π‘”) / π‘ β¦Œ{π‘Ž ∈ Xπ‘₯ ∈ (Baseβ€˜πΆ)((π‘Ÿβ€˜π‘₯)(Hom β€˜π·)(π‘ β€˜π‘₯)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜πΆ)βˆ€π‘¦ ∈ (Baseβ€˜πΆ)βˆ€π‘§ ∈ (π‘₯(Hom β€˜πΆ)𝑦)((π‘Žβ€˜π‘¦)(⟨(π‘Ÿβ€˜π‘₯), (π‘Ÿβ€˜π‘¦)⟩(compβ€˜π·)(π‘ β€˜π‘¦))((π‘₯(2nd β€˜π‘“)𝑦)β€˜π‘§)) = (((π‘₯(2nd β€˜π‘”)𝑦)β€˜π‘§)(⟨(π‘Ÿβ€˜π‘₯), (π‘ β€˜π‘₯)⟩(compβ€˜π·)(π‘ β€˜π‘¦))(π‘Žβ€˜π‘₯))} ∈ 𝒫 (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)) ↔ (𝐢 Nat 𝐷):((𝐢 Func 𝐷) Γ— (𝐢 Func 𝐷))βŸΆπ’« (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
4638, 45mpbi 229 . . 3 (𝐢 Nat 𝐷):((𝐢 Func 𝐷) Γ— (𝐢 Func 𝐷))βŸΆπ’« (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ))
4746a1i 11 . 2 (πœ‘ β†’ (𝐢 Nat 𝐷):((𝐢 Func 𝐷) Γ— (𝐢 Func 𝐷))βŸΆπ’« (βˆͺ ran (Hom β€˜π·) ↑m (Baseβ€˜πΆ)))
481, 5, 13, 47wunf 10719 1 (πœ‘ β†’ (𝐢 Nat 𝐷) ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475  [wsbc 3777  β¦‹csb 3893   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βˆͺ cuni 4908   Γ— cxp 5674  ran crn 5677  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971   ↑m cmap 8817  Xcixp 8888  WUnicwun 10692  ndxcnx 17123  Basecbs 17141  Hom chom 17205  compcco 17206   Func cfunc 17801   Nat cnat 17889
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-map 8819  df-pm 8820  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-wun 10694  df-pnf 11247  df-mnf 11248  df-ltxr 11250  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-dec 12675  df-slot 17112  df-ndx 17124  df-base 17142  df-hom 17218  df-func 17805  df-nat 17891
This theorem is referenced by:  catcfuccl  18066  catcfucclOLD  18067
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