Theorem wunfunc 17996

Description: A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)

Hypotheses

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

Assertion

Ref	Expression
wunfunc	⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Proof of Theorem wunfunc

Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wunfunc.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		baseid 17310	. . . . 5 ⊢ Base = Slot (Base‘ndx)
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 17286	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 17286	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 10739	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
8		homid 17503	. . . . . . . . 9 ⊢ Hom = Slot (Hom ‘ndx)
9	8, 1, 5	wunstr 17286	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 10742	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 10719	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 17286	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 10742	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 10719	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 10737	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 10720	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 10737	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 10739	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 10737	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 17957	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 5108	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 17961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6895	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6895	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8882	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)))
31		mapsspw 8889	. . . . . . . . . . 11 ⊢ (((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))))
32		fvssunirn 6913	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 7453	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5674	. . . . . . . . . . . . 13 ⊢ ((((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶) ∧ ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)) → (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)))
35	32, 33, 34	mp2an 705	. . . . . . . . . . . 12 ⊢ (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
36	35	sspwi 4572	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
37	31, 36	sstri 3943	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	37	rgenw 3082	. . . . . . . . 9 ⊢ ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
39		ss2ixp 8921	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) → X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)))
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
41	28, 28	xpex 7756	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42		fvex 6895	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
43	42	rnex 7911	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
44	43	uniex 7747	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
45		fvex 6895	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
46	45	rnex 7911	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
47	46	uniex 7747	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
48	44, 47	xpex 7756	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
49	48	pwex 5349	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	41, 49	ixpconst 8918	. . . . . . . 8 ⊢ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) = (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))
51	40, 50	sseqtri 3982	. . . . . . 7 ⊢ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))
52		eqid 2762	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		eqid 2762	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
54	23, 52, 53, 25	funcixp 17962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)))
55	51, 54	sselid 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))
56	30, 55	opelxpd 5698	. . . . 5 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → ⟨𝑓, 𝑔⟩ ∈ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))))
57	56	ex 418	. . . 4 ⊢ (𝜑 → (𝑓(𝐶 Func 𝐷)𝑔 → ⟨𝑓, 𝑔⟩ ∈ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))))
58	22, 57	biimtrrid 246	. . 3 ⊢ (𝜑 → (⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷) → ⟨𝑓, 𝑔⟩ ∈ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))))
59	21, 58	relssdv 5772	. 2 ⊢ (𝜑 → (𝐶 Func 𝐷) ⊆ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))))
60	1, 19, 59	wunss 10725	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  𝒫 cpw 4560  ⟨cop 4593  ∪ cuni 4870   class class class wbr 5107   × cxp 5657  ran crn 5660  Rel wrel 5664  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8830  Xcixp 8908  WUnicwun 10713  ndxcnx 17291  Basecbs 17307  Hom chom 17359   Func cfunc 17949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-wun 10715  df-pnf 11273  df-mnf 11274  df-ltxr 11276  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-dec 12741  df-slot 17280  df-ndx 17292  df-base 17308  df-hom 17372  df-func 17953

This theorem is used by:  wunnat  18054  catcfuccl  18213