| Step | Hyp | Ref
| Expression |
| 1 | | catcfuccl.o |
. . . . 5
⊢ 𝑄 = (𝑋 FuncCat 𝑌) |
| 2 | | eqid 2737 |
. . . . 5
⊢ (𝑋 Func 𝑌) = (𝑋 Func 𝑌) |
| 3 | | eqid 2737 |
. . . . 5
⊢ (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌) |
| 4 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑋) =
(Base‘𝑋) |
| 5 | | eqid 2737 |
. . . . 5
⊢
(comp‘𝑌) =
(comp‘𝑌) |
| 6 | | catcfuccl.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 7 | | catcfuccl.c |
. . . . . . . 8
⊢ 𝐶 = (CatCat‘𝑈) |
| 8 | | catcfuccl.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐶) |
| 9 | | catcfuccl.u |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ WUni) |
| 10 | 7, 8, 9 | catcbas 18146 |
. . . . . . 7
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
| 11 | 6, 10 | eleqtrd 2843 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat)) |
| 12 | 11 | elin2d 4205 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ Cat) |
| 13 | | catcfuccl.y |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 14 | 13, 10 | eleqtrd 2843 |
. . . . . 6
⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat)) |
| 15 | 14 | elin2d 4205 |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ Cat) |
| 16 | | eqidd 2738 |
. . . . 5
⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))) |
| 17 | 1, 2, 3, 4, 5, 12,
15, 16 | fucval 18006 |
. . . 4
⊢ (𝜑 → 𝑄 = {〈(Base‘ndx), (𝑋 Func 𝑌)〉, 〈(Hom ‘ndx), (𝑋 Nat 𝑌)〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉}) |
| 18 | | baseid 17250 |
. . . . . . 7
⊢ Base =
Slot (Base‘ndx) |
| 19 | | catcfuccl.1 |
. . . . . . . 8
⊢ (𝜑 → ω ∈ 𝑈) |
| 20 | 9, 19 | wunndx 17232 |
. . . . . . 7
⊢ (𝜑 → ndx ∈ 𝑈) |
| 21 | 18, 9, 20 | wunstr 17225 |
. . . . . 6
⊢ (𝜑 → (Base‘ndx) ∈
𝑈) |
| 22 | 7, 8, 9, 6 | catcbascl 18157 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝑈) |
| 23 | 7, 8, 9, 13 | catcbascl 18157 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ 𝑈) |
| 24 | 9, 22, 23 | wunfunc 17946 |
. . . . . 6
⊢ (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈) |
| 25 | 9, 21, 24 | wunop 10762 |
. . . . 5
⊢ (𝜑 → 〈(Base‘ndx),
(𝑋 Func 𝑌)〉 ∈ 𝑈) |
| 26 | | homid 17456 |
. . . . . . 7
⊢ Hom =
Slot (Hom ‘ndx) |
| 27 | 26, 9, 20 | wunstr 17225 |
. . . . . 6
⊢ (𝜑 → (Hom ‘ndx) ∈
𝑈) |
| 28 | 9, 22, 23 | wunnat 18004 |
. . . . . 6
⊢ (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈) |
| 29 | 9, 27, 28 | wunop 10762 |
. . . . 5
⊢ (𝜑 → 〈(Hom ‘ndx),
(𝑋 Nat 𝑌)〉 ∈ 𝑈) |
| 30 | | ccoid 17458 |
. . . . . . 7
⊢ comp =
Slot (comp‘ndx) |
| 31 | 30, 9, 20 | wunstr 17225 |
. . . . . 6
⊢ (𝜑 → (comp‘ndx) ∈
𝑈) |
| 32 | 9, 24, 24 | wunxp 10764 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈) |
| 33 | 9, 32, 24 | wunxp 10764 |
. . . . . . 7
⊢ (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈) |
| 34 | 7, 8, 9, 13 | catcccocl 18161 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈) |
| 35 | 9, 34 | wunrn 10769 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈) |
| 36 | 9, 35 | wununi 10746 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈) |
| 37 | 9, 36 | wunrn 10769 |
. . . . . . . . . . 11
⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
| 38 | 9, 37 | wununi 10746 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
| 39 | 9, 38 | wunpw 10747 |
. . . . . . . . 9
⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
| 40 | 7, 8, 9, 6 | catcbaselcl 18159 |
. . . . . . . . 9
⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈) |
| 41 | 9, 39, 40 | wunmap 10766 |
. . . . . . . 8
⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) ∈
𝑈) |
| 42 | 9, 28 | wunrn 10769 |
. . . . . . . . . 10
⊢ (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈) |
| 43 | 9, 42 | wununi 10746 |
. . . . . . . . 9
⊢ (𝜑 → ∪ ran (𝑋 Nat 𝑌) ∈ 𝑈) |
| 44 | 9, 43, 43 | wunxp 10764 |
. . . . . . . 8
⊢ (𝜑 → (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)) ∈ 𝑈) |
| 45 | 9, 41, 44 | wunpm 10765 |
. . . . . . 7
⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) ∈ 𝑈) |
| 46 | | fvex 6919 |
. . . . . . . . . . 11
⊢
(1st ‘𝑣) ∈ V |
| 47 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘𝑣) ∈ V |
| 48 | | ovex 7464 |
. . . . . . . . . . . . . . . . 17
⊢
(𝒫 ∪ ran ∪
ran (comp‘𝑌)
↑m (Base‘𝑋)) ∈ V |
| 49 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 Nat 𝑌) ∈ V |
| 50 | 49 | rnex 7932 |
. . . . . . . . . . . . . . . . . . 19
⊢ ran
(𝑋 Nat 𝑌) ∈ V |
| 51 | 50 | uniex 7761 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ran (𝑋 Nat 𝑌) ∈ V |
| 52 | 51, 51 | xpex 7773 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)) ∈ V |
| 53 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) |
| 54 | | ovssunirn 7467 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran
(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) |
| 55 | | ovssunirn 7467 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ∪ ran
(comp‘𝑌) |
| 56 | | rnss 5950 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ∪ ran
(comp‘𝑌) → ran
(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ran ∪
ran (comp‘𝑌)) |
| 57 | | uniss 4915 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ran
(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ran ∪
ran (comp‘𝑌) →
∪ ran (〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ∪ ran
∪ ran (comp‘𝑌)) |
| 58 | 55, 56, 57 | mp2b 10 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ ran (〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥)) ⊆ ∪ ran
∪ ran (comp‘𝑌) |
| 59 | 54, 58 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran
∪ ran (comp‘𝑌) |
| 60 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ V |
| 61 | 60 | elpw 4604 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ↔ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran
∪ ran (comp‘𝑌)) |
| 62 | 59, 61 | mpbir 231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌) |
| 63 | 62 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (Base‘𝑋) → ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)) |
| 64 | 53, 63 | fmpti 7132 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))):(Base‘𝑋)⟶𝒫 ∪ ran ∪ ran (comp‘𝑌) |
| 65 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(comp‘𝑌)
∈ V |
| 66 | 65 | rnex 7932 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ran
(comp‘𝑌) ∈
V |
| 67 | 66 | uniex 7761 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ ran (comp‘𝑌) ∈ V |
| 68 | 67 | rnex 7932 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran ∪ ran (comp‘𝑌) ∈ V |
| 69 | 68 | uniex 7761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V |
| 70 | 69 | pwex 5380 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝒫
∪ ran ∪ ran
(comp‘𝑌) ∈
V |
| 71 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(Base‘𝑋)
∈ V |
| 72 | 70, 71 | elmap 8911 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) ↔
(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))):(Base‘𝑋)⟶𝒫 ∪ ran ∪ ran (comp‘𝑌)) |
| 73 | 64, 72 | mpbir 231 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) |
| 74 | 73 | rgen2w 3066 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑏 ∈
(𝑔(𝑋 Nat 𝑌)ℎ)∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) |
| 75 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) |
| 76 | 75 | fmpo 8093 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑏 ∈
(𝑔(𝑋 Nat 𝑌)ℎ)∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) ↔
(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))) |
| 77 | 74, 76 | mpbi 230 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) |
| 78 | | ovssunirn 7467 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran
(𝑋 Nat 𝑌) |
| 79 | | ovssunirn 7467 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran
(𝑋 Nat 𝑌) |
| 80 | | xpss12 5700 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran
(𝑋 Nat 𝑌) ∧ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran
(𝑋 Nat 𝑌)) → ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran
(𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 81 | 78, 79, 80 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran
(𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)) |
| 82 | | elpm2r 8885 |
. . . . . . . . . . . . . . . . 17
⊢
((((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m (Base‘𝑋)) ∈ V ∧ (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)) ∈ V) ∧ ((𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋)) ∧
((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran
(𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) → (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 83 | 48, 52, 77, 81, 82 | mp4an 693 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 84 | 83 | sbcth 3803 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘𝑣) ∈ V → [(2nd
‘𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 85 | | sbcel1g 4416 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘𝑣) ∈ V → ([(2nd
‘𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) ↔ ⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))))) |
| 86 | 84, 85 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘𝑣) ∈ V →
⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 87 | 47, 86 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 88 | 87 | sbcth 3803 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑣) ∈ V → [(1st
‘𝑣) / 𝑓]⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 89 | | sbcel1g 4416 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑣) ∈ V → ([(1st
‘𝑣) / 𝑓]⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) ↔ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))))) |
| 90 | 88, 89 | mpbid 232 |
. . . . . . . . . . 11
⊢
((1st ‘𝑣) ∈ V →
⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 91 | 46, 90 | ax-mp 5 |
. . . . . . . . . 10
⊢
⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 92 | 91 | rgen2w 3066 |
. . . . . . . . 9
⊢
∀𝑣 ∈
((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ℎ ∈ (𝑋 Func 𝑌)⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 93 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) |
| 94 | 93 | fmpo 8093 |
. . . . . . . . 9
⊢
(∀𝑣 ∈
((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ℎ ∈ (𝑋 Func 𝑌)⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) ↔ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 95 | 92, 94 | mpbi 230 |
. . . . . . . 8
⊢ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌))) |
| 96 | 95 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑m
(Base‘𝑋))
↑pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran
(𝑋 Nat 𝑌)))) |
| 97 | 9, 33, 45, 96 | wunf 10767 |
. . . . . 6
⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) ∈ 𝑈) |
| 98 | 9, 31, 97 | wunop 10762 |
. . . . 5
⊢ (𝜑 → 〈(comp‘ndx),
(𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉 ∈ 𝑈) |
| 99 | 9, 25, 29, 98 | wuntp 10751 |
. . . 4
⊢ (𝜑 → {〈(Base‘ndx),
(𝑋 Func 𝑌)〉, 〈(Hom ‘ndx), (𝑋 Nat 𝑌)〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑌)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉} ∈ 𝑈) |
| 100 | 17, 99 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → 𝑄 ∈ 𝑈) |
| 101 | 1, 12, 15 | fuccat 18018 |
. . 3
⊢ (𝜑 → 𝑄 ∈ Cat) |
| 102 | 100, 101 | elind 4200 |
. 2
⊢ (𝜑 → 𝑄 ∈ (𝑈 ∩ Cat)) |
| 103 | 102, 10 | eleqtrrd 2844 |
1
⊢ (𝜑 → 𝑄 ∈ 𝐵) |