wunfunc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wunfunc		Structured version Visualization version GIF version

Theorem wunfunc 16918

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 16235	. . . . 5 ⊢ Base = Slot 1
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 16253	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 16253	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 9870	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑_𝑚 (Base‘𝐶)) ∈ 𝑈)
8		df-hom 16336	. . . . . . . . 9 ⊢ Hom = Slot ;14
9	8, 1, 5	wunstr 16253	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 9873	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 9850	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 16253	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 9873	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 9850	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 9868	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 9851	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 9868	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 9870	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_𝑚 ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 9868	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑_𝑚 (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_𝑚 ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 16881	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 4876	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2825	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2825	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 16885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6450	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6450	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8156	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑_𝑚 (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑_𝑚 (Base‘𝐶)))
31		mapsspw 8163	. . . . . . . . . . 11 ⊢ (((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))))
32		fvssunirn 6466	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 6945	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5361	. . . . . . . . . . . . 13 ⊢ ((((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶) ∧ ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)) → (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)))
35	32, 33, 34	mp2an 683	. . . . . . . . . . . 12 ⊢ (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
36		sspwb 5140	. . . . . . . . . . . 12 ⊢ ((((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↔ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)))
37	35, 36	mpbi 222	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	31, 37	sstri 3836	. . . . . . . . . 10 ⊢ (((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
39	38	rgenw 3133	. . . . . . . . 9 ⊢ ∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
40		ss2ixp 8194	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) → X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘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‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
42	28, 28	xpex 7228	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
43		fvex 6450	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
44	43	rnex 7367	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
45	44	uniex 7218	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
46		fvex 6450	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
47	46	rnex 7367	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
48	47	uniex 7218	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
49	45, 48	xpex 7228	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	49	pwex 5082	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
51	42, 50	ixpconst 8191	. . . . . . . 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 3862	. . . . . . 7 ⊢ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_𝑚 ((Base‘𝐶) × (Base‘𝐶)))
53		eqid 2825	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
54		eqid 2825	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
55	23, 53, 54, 25	funcixp 16886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_𝑚 ((Hom ‘𝐶)‘𝑧)))
56	52, 55	sseldi 3825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_𝑚 ((Base‘𝐶) × (Base‘𝐶))))
57		opelxpi 5383	. . . . . 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 579	. . . . 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 403	. . . 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 235	. . 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 5450	. 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 9856	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ∀wral 3117 ⊆ wss 3798 𝒫 cpw 4380 ⟨cop 4405 ∪ cuni 4660 class class class wbr 4875 × cxp 5344 ran crn 5347 Rel wrel 5351 ⟶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 Func cfunc 16873

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-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

This theorem is referenced by: wunnat 16975 catcfuccl 17118

W3C validator