wunfunc - Metamath Proof Explorer

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Theorem wunfunc 17171

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 16491	. . . . 5 ⊢ Base = Slot 1
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 16509	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 16509	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 10150	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
8		df-hom 16591	. . . . . . . . 9 ⊢ Hom = Slot ;14
9	8, 1, 5	wunstr 16509	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 10153	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 10130	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 16509	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 10153	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 10130	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 10148	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 10131	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 10148	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 10150	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 10148	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑_m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 17134	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 5069	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2823	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2823	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 17138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6685	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6685	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8437	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑_m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑_m (Base‘𝐶)))
31		mapsspw 8444	. . . . . . . . . . 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 6701	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 7194	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5572	. . . . . . . . . . . . 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 690	. . . . . . . . . . . 12 ⊢ (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
36	35	sspwi 4555	. . . . . . . . . . 11 ⊢ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧)))) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
37	31, 36	sstri 3978	. . . . . . . . . 10 ⊢ (((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ↑_m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷))
38	37	rgenw 3152	. . . . . . . . 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))
39		ss2ixp 8476	. . . . . . . . 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)))
40	38, 39	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))
41	28, 28	xpex 7478	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42		fvex 6685	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
43	42	rnex 7619	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
44	43	uniex 7469	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
45		fvex 6685	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
46	45	rnex 7619	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
47	46	uniex 7469	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
48	44, 47	xpex 7478	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
49	48	pwex 5283	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	41, 49	ixpconst 8473	. . . . . . . 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 4005	. . . . . . 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)))
52		eqid 2823	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		eqid 2823	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
54	23, 52, 53, 25	funcixp 17139	. . . . . . 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)))
55	51, 54	sseldi 3967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶))))
56	30, 55	opelxpd 5595	. . . . 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 415	. . . 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	syl5bir 245	. . 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 5663	. 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 10136	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 𝒫 cpw 4541 ⟨cop 4575 ∪ cuni 4840 class class class wbr 5068 × cxp 5555 ran crn 5558 Rel wrel 5562 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 ↑_m cmap 8408 Xcixp 8463 WUnicwun 10124 1c1 10540 4c4 11697 cdc 12101 Basecbs 16485 Hom chom 16578 Func cfunc 17126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-pm 8411 df-ixp 8464 df-wun 10126 df-slot 16489 df-base 16491 df-hom 16591 df-func 17130

This theorem is referenced by: wunnat 17228 catcfuccl 17371

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