wunfunc - Metamath Proof Explorer

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Theorem wunfunc 17853

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 17151	. . . . 5 ⊢ Base = Slot (Base‘ndx)
3		wunfunc.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
4	2, 1, 3	wunstr 17125	. . . 4 ⊢ (𝜑 → (Base‘𝐷) ∈ 𝑈)
5		wunfunc.2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
6	2, 1, 5	wunstr 17125	. . . 4 ⊢ (𝜑 → (Base‘𝐶) ∈ 𝑈)
7	1, 4, 6	wunmap 10723	. . 3 ⊢ (𝜑 → ((Base‘𝐷) ↑_m (Base‘𝐶)) ∈ 𝑈)
8		homid 17361	. . . . . . . . 9 ⊢ Hom = Slot (Hom ‘ndx)
9	8, 1, 5	wunstr 17125	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
10	1, 9	wunrn 10726	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
11	1, 10	wununi 10703	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐶) ∈ 𝑈)
12	8, 1, 3	wunstr 17125	. . . . . . . 8 ⊢ (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
13	1, 12	wunrn 10726	. . . . . . 7 ⊢ (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
14	1, 13	wununi 10703	. . . . . 6 ⊢ (𝜑 → ∪ ran (Hom ‘𝐷) ∈ 𝑈)
15	1, 11, 14	wunxp 10721	. . . . 5 ⊢ (𝜑 → (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
16	1, 15	wunpw 10704	. . . 4 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ 𝑈)
17	1, 6, 6	wunxp 10721	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
18	1, 16, 17	wunmap 10723	. . 3 ⊢ (𝜑 → (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
19	1, 7, 18	wunxp 10721	. 2 ⊢ (𝜑 → (((Base‘𝐷) ↑_m (Base‘𝐶)) × (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20		relfunc 17816	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
21	20	a1i 11	. . 3 ⊢ (𝜑 → Rel (𝐶 Func 𝐷))
22		df-br 5149	. . . 4 ⊢ (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
24		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
25		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
26	23, 24, 25	funcf1 17820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27		fvex 6904	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
28		fvex 6904	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
29	27, 28	elmap 8867	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘𝐷) ↑_m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
30	26, 29	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑_m (Base‘𝐶)))
31		mapsspw 8874	. . . . . . . . . . 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 6924	. . . . . . . . . . . . 13 ⊢ ((Hom ‘𝐶)‘𝑧) ⊆ ∪ ran (Hom ‘𝐶)
33		ovssunirn 7447	. . . . . . . . . . . . 13 ⊢ ((𝑓‘(1^st ‘𝑧))(Hom ‘𝐷)(𝑓‘(2^nd ‘𝑧))) ⊆ ∪ ran (Hom ‘𝐷)
34		xpss12 5691	. . . . . . . . . . . . 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 4614	. . . . . . . . . . 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 3991	. . . . . . . . . 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 3065	. . . . . . . . 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 8906	. . . . . . . . 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 7742	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42		fvex 6904	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) ∈ V
43	42	rnex 7905	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐶) ∈ V
44	43	uniex 7733	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐶) ∈ V
45		fvex 6904	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) ∈ V
46	45	rnex 7905	. . . . . . . . . . . 12 ⊢ ran (Hom ‘𝐷) ∈ V
47	46	uniex 7733	. . . . . . . . . . 11 ⊢ ∪ ran (Hom ‘𝐷) ∈ V
48	44, 47	xpex 7742	. . . . . . . . . 10 ⊢ (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
49	48	pwex 5378	. . . . . . . . 9 ⊢ 𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ∈ V
50	41, 49	ixpconst 8903	. . . . . . . 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 4018	. . . . . . 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 2732	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
53		eqid 2732	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
54	23, 52, 53, 25	funcixp 17821	. . . . . . 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	sselid 3980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 (∪ ran (Hom ‘𝐶) × ∪ ran (Hom ‘𝐷)) ↑_m ((Base‘𝐶) × (Base‘𝐶))))
56	30, 55	opelxpd 5715	. . . . 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 413	. . . 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 242	. . 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 5788	. 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 10709	1 ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 𝒫 cpw 4602 ⟨cop 4634 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ran crn 5677 Rel wrel 5681 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 Xcixp 8893 WUnicwun 10697 ndxcnx 17130 Basecbs 17148 Hom chom 17212 Func cfunc 17808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-wun 10699 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-dec 12682 df-slot 17119 df-ndx 17131 df-base 17149 df-hom 17225 df-func 17812

This theorem is referenced by: wunnat 17911 wunnatOLD 17912 catcfuccl 18073 catcfucclOLD 18074

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