Theorem wunfunc 17826

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 17141	. . . . 5 ⊢ Base = Slot (Base‘ndx)
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 17117	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 17117	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 10639	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
8		homid 17334	. . . . . . . . 9 ⊢ Hom = Slot (Hom ‘ndx)
9	8, 1, 5	wunstr 17117	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 10642	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 10619	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 17117	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 10642	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 10619	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 10637	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 10620	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 10637	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 10639	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 10637	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 17787	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 5096	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 17791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6839	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6839	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8805	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)))
31		mapsspw 8812	. . . . . . . . . . 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 6857	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 7389	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5638	. . . . . . . . . . . . 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 692	. . . . . . . . . . . 12 ⊢ (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
36	35	sspwi 4565	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
37	31, 36	sstri 3947	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	37	rgenw 3048	. . . . . . . . 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 8844	. . . . . . . . 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 7693	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42		fvex 6839	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
43	42	rnex 7850	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
44	43	uniex 7681	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
45		fvex 6839	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
46	45	rnex 7850	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
47	46	uniex 7681	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
48	44, 47	xpex 7693	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
49	48	pwex 5322	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	41, 49	ixpconst 8841	. . . . . . . 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 3986	. . . . . . 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 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
54	23, 52, 53, 25	funcixp 17792	. . . . . . 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 3935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))
56	30, 55	opelxpd 5662	. . . . 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 412	. . . 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 243	. . 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 5735	. 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 10625	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905  𝒫 cpw 4553  ⟨cop 4585  ∪ cuni 4861   class class class wbr 5095   × cxp 5621  ran crn 5624  Rel wrel 5628  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930   ↑_m cmap 8760  Xcixp 8831  WUnicwun 10613  ndxcnx 17122  Basecbs 17138  Hom chom 17190   Func cfunc 17779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-pm 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-wun 10615  df-pnf 11170  df-mnf 11171  df-ltxr 11173  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-dec 12610  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-func 17783

This theorem is referenced by:  wunnat  17884  catcfuccl  18043