Theorem wunfunc 17983

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 17297	. . . . 5 ⊢ Base = Slot (Base‘ndx)
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 17273	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 17273	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 10729	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
8		homid 17490	. . . . . . . . 9 ⊢ Hom = Slot (Hom ‘ndx)
9	8, 1, 5	wunstr 17273	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 10732	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 10709	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 17273	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 10732	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 10709	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 10727	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 10710	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 10727	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 10729	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 10727	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 17944	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 5115	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2766	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2766	. . . . . . . 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 17948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6901	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6901	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8878	. . . . . . 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 8885	. . . . . . . . . . 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 6919	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 7459	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5681	. . . . . . . . . . . . 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 4579	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
37	31, 36	sstri 3949	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	37	rgenw 3086	. . . . . . . . 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 8917	. . . . . . . . 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 7761	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42		fvex 6901	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
43	42	rnex 7916	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
44	43	uniex 7752	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
45		fvex 6901	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
46	45	rnex 7916	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
47	46	uniex 7752	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
48	44, 47	xpex 7761	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
49	48	pwex 5356	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	41, 49	ixpconst 8914	. . . . . . . 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 3988	. . . . . . 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 2766	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		eqid 2766	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
54	23, 52, 53, 25	funcixp 17949	. . . . . . 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 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))
56	30, 55	opelxpd 5705	. . . . 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 5779	. 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 10715	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908  𝒫 cpw 4567  ⟨cop 4600  ∪ cuni 4877   class class class wbr 5114   × cxp 5664  ran crn 5667  Rel wrel 5671  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994   ↑_m cmap 8833  Xcixp 8904  WUnicwun 10703  ndxcnx 17278  Basecbs 17294  Hom chom 17346   Func cfunc 17936

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-wun 10705  df-pnf 11263  df-mnf 11264  df-ltxr 11266  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-dec 12730  df-slot 17267  df-ndx 17279  df-base 17295  df-hom 17359  df-func 17940

This theorem is used by:  wunnat  18041  catcfuccl  18200