Theorem wunnat 16975

Description: A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wunnat.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunnat.2	⊢ (𝜑 → 𝐶 ∈ 𝑈)
wunnat.3	⊢ (𝜑 → 𝐷 ∈ 𝑈)

Assertion

Ref	Expression
wunnat	⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Proof of Theorem wunnat

Dummy variables 𝑓 𝑎 𝑔 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wunnat.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunnat.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
3		wunnat.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	1, 2, 3	wunfunc 16918	. . 3 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
5	1, 4, 4	wunxp 9868	. 2 ⊢ (𝜑 → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ∈ 𝑈)
6		df-hom 16336	. . . . . . 7 ⊢ Hom = Slot ;14
7	6, 1, 3	wunstr 16253	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
8	1, 7	wunrn 9873	. . . . 5 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
9	1, 8	wununi 9850	. . . 4 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
10		df-base 16235	. . . . 5 ⊢ Base = Slot 1
11	10, 1, 2	wunstr 16253	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
12	1, 9, 11	wunmap 9870	. . 3 ⊢ (𝜑 → (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
13	1, 12	wunpw 9851	. 2 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
14		fvex 6450	. . . . . 6 ⊢ (1st ‘𝑓) ∈ V
15		fvex 6450	. . . . . . . . 9 ⊢ (1st ‘𝑔) ∈ V
16		ssrab2 3914	. . . . . . . . . . . . 13 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥))
17		ovssunirn 6945	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
18	17	rgenw 3133	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷)
19		ss2ixp 8194	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ ∪ ran (Hom ‘𝐷) → X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷))
20	18, 19	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷)
21		fvex 6450	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐶) ∈ V
22		fvex 6450	. . . . . . . . . . . . . . . . 17 ⊢ (Hom ‘𝐷) ∈ V
23	22	rnex 7367	. . . . . . . . . . . . . . . 16 ⊢ ran (Hom ‘𝐷) ∈ V
24	23	uniex 7218	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
25	21, 24	ixpconst 8191	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (Base‘𝐶)∪ ran (Hom ‘𝐷) = (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
26	20, 25	sseqtri 3862	. . . . . . . . . . . . 13 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ⊆ (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
27	16, 26	sstri 3836	. . . . . . . . . . . 12 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
28		ovex 6942	. . . . . . . . . . . . 13 ⊢ (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ V
29	28	elpw2 5052	. . . . . . . . . . . 12 ⊢ ({𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ⊆ (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
30	27, 29	mpbir 223	. . . . . . . . . . 11 ⊢ {𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
31	30	sbcth 3677	. . . . . . . . . 10 ⊢ ((1st ‘𝑔) ∈ V → [(1st ‘𝑔) / 𝑠]{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
32		sbcel1g 4213	. . . . . . . . . 10 ⊢ ((1st ‘𝑔) ∈ V → ([(1st ‘𝑔) / 𝑠]{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))))
33	31, 32	mpbid 224	. . . . . . . . 9 ⊢ ((1st ‘𝑔) ∈ V → ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
34	15, 33	ax-mp 5	. . . . . . . 8 ⊢ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
35	34	sbcth 3677	. . . . . . 7 ⊢ ((1st ‘𝑓) ∈ V → [(1st ‘𝑓) / 𝑟]⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
36		sbcel1g 4213	. . . . . . 7 ⊢ ((1st ‘𝑓) ∈ V → ([(1st ‘𝑓) / 𝑟]⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))))
37	35, 36	mpbid 224	. . . . . 6 ⊢ ((1st ‘𝑓) ∈ V → ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
38	14, 37	ax-mp 5	. . . . 5 ⊢ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
39	38	rgen2w 3134	. . . 4 ⊢ ∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
40		eqid 2825	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
41		eqid 2825	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
42		eqid 2825	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
43		eqid 2825	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
44		eqid 2825	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
45	40, 41, 42, 43, 44	natfval 16965	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
46	45	fmpt2 7505	. . . 4 ⊢ (∀𝑓 ∈ (𝐶 Func 𝐷)∀𝑔 ∈ (𝐶 Func 𝐷)⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘𝑧)) = (((𝑥(2nd ‘𝑔)𝑦)‘𝑧)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))} ∈ 𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
47	39, 46	mpbi 222	. . 3 ⊢ (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶))
48	47	a1i 11	. 2 ⊢ (𝜑 → (𝐶 Nat 𝐷):((𝐶 Func 𝐷) × (𝐶 Func 𝐷))⟶𝒫 (∪ ran (Hom ‘𝐷) ↑𝑚 (Base‘𝐶)))
49	1, 5, 13, 48	wunf 9871	1 ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ∀wral 3117  {crab 3121  Vcvv 3414  [wsbc 3662  ⦋csb 3757   ⊆ wss 3798  𝒫 cpw 4380  ⟨cop 4405  ∪ cuni 4660   × cxp 5344  ran crn 5347  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  1^st c1st 7431  2^nd c2nd 7432   ↑_𝑚 cmap 8127  Xcixp 8181  WUnicwun 9844  1c1 10260  4c4 11415  ;cdc 11828  Basecbs 16229  Hom chom 16323  compcco 16324   Func cfunc 16873   Nat cnat 16960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-pm 8130  df-ixp 8182  df-wun 9846  df-slot 16233  df-base 16235  df-hom 16336  df-func 16877  df-nat 16962

This theorem is referenced by:  catcfuccl  17118