Theorem wunfunc 16559

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

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		df-base 15863	. . . . 5 ⊢ Base = Slot 1
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 15881	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 15881	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 9548	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
8		df-hom 15966	. . . . . . . . 9 ⊢ Hom = Slot ;14
9	8, 1, 5	wunstr 15881	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 9551	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 9528	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 15881	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 9551	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 9528	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 9546	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 9529	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 9546	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 9548	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 9546	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 16522	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 4654	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 16526	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 7886	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)))
31		mapsspw 7893	. . . . . . . . . . 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 6217	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 6681	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5225	. . . . . . . . . . . . 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 708	. . . . . . . . . . . 12 ⊢ (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
36		sspwb 4917	. . . . . . . . . . . 12 ⊢ ((((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↔ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)))
37	35, 36	mpbi 220	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	31, 37	sstri 3612	. . . . . . . . . 10 ⊢ (((𝑓‘(1st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
39	38	rgenw 2924	. . . . . . . . 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))
40		ss2ixp 7921	. . . . . . . . 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)))
41	39, 40	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))
42	28, 28	xpex 6962	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
43		fvex 6201	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
44	43	rnex 7100	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
45	44	uniex 6953	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
46		fvex 6201	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
47	46	rnex 7100	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
48	47	uniex 6953	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
49	45, 48	xpex 6962	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	49	pwex 4848	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
51	42, 50	ixpconst 7918	. . . . . . . 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)))
52	41, 51	sseqtri 3637	. . . . . . 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)))
53		eqid 2622	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
54		eqid 2622	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
55	23, 53, 54, 25	funcixp 16527	. . . . . . 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)))
56	52, 55	sseldi 3601	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))))
57		opelxpi 5148	. . . . . 6 ⊢ ((𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ∧ 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))) → ⟨𝑓, 𝑔⟩ ∈ (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))))
58	30, 56, 57	syl2anc 693	. . . . 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)))))
59	58	ex 450	. . . 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))))))
60	22, 59	syl5bir 233	. . 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))))))
61	21, 60	relssdv 5212	. 2 ⊢ (𝜑 → (𝐶 Func 𝐷) ⊆ (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))))
62	1, 19, 61	wunss 9534	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   × cxp 5112  ran crn 5115  Rel wrel 5119  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Xcixp 7908  WUnicwun 9522  1c1 9937  4c4 11072  ;cdc 11493  Basecbs 15857  Hom chom 15952   Func cfunc 16514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-pm 7860  df-ixp 7909  df-wun 9524  df-slot 15861  df-base 15863  df-hom 15966  df-func 16518

This theorem is referenced by:  wunnat  16616  catcfuccl  16759