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Theorem wunfunc 17946
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 (𝜑𝑈 ∈ WUni)
wunfunc.2 (𝜑𝐶𝑈)
wunfunc.3 (𝜑𝐷𝑈)
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.
StepHypRef Expression
1 wunfunc.1 . 2 (𝜑𝑈 ∈ WUni)
2 baseid 17250 . . . . 5 Base = Slot (Base‘ndx)
3 wunfunc.3 . . . . 5 (𝜑𝐷𝑈)
42, 1, 3wunstr 17225 . . . 4 (𝜑 → (Base‘𝐷) ∈ 𝑈)
5 wunfunc.2 . . . . 5 (𝜑𝐶𝑈)
62, 1, 5wunstr 17225 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
71, 4, 6wunmap 10766 . . 3 (𝜑 → ((Base‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
8 homid 17456 . . . . . . . . 9 Hom = Slot (Hom ‘ndx)
98, 1, 5wunstr 17225 . . . . . . . 8 (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
101, 9wunrn 10769 . . . . . . 7 (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
111, 10wununi 10746 . . . . . 6 (𝜑 ran (Hom ‘𝐶) ∈ 𝑈)
128, 1, 3wunstr 17225 . . . . . . . 8 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
131, 12wunrn 10769 . . . . . . 7 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
141, 13wununi 10746 . . . . . 6 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
151, 11, 14wunxp 10764 . . . . 5 (𝜑 → ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
161, 15wunpw 10747 . . . 4 (𝜑 → 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
171, 6, 6wunxp 10764 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
181, 16, 17wunmap 10766 . . 3 (𝜑 → (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
191, 7, 18wunxp 10764 . 2 (𝜑 → (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20 relfunc 17907 . . . 4 Rel (𝐶 Func 𝐷)
2120a1i 11 . . 3 (𝜑 → Rel (𝐶 Func 𝐷))
22 df-br 5144 . . . 4 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23 eqid 2737 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
24 eqid 2737 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
25 simpr 484 . . . . . . . 8 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
2623, 24, 25funcf1 17911 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27 fvex 6919 . . . . . . . 8 (Base‘𝐷) ∈ V
28 fvex 6919 . . . . . . . 8 (Base‘𝐶) ∈ V
2927, 28elmap 8911 . . . . . . 7 (𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
3026, 29sylibr 234 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)))
31 mapsspw 8918 . . . . . . . . . . 11 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))))
32 fvssunirn 6939 . . . . . . . . . . . . 13 ((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶)
33 ovssunirn 7467 . . . . . . . . . . . . 13 ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)
34 xpss12 5700 . . . . . . . . . . . . 13 ((((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶) ∧ ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)) → (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)))
3532, 33, 34mp2an 692 . . . . . . . . . . . 12 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3635sspwi 4612 . . . . . . . . . . 11 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3731, 36sstri 3993 . . . . . . . . . 10 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3837rgenw 3065 . . . . . . . . 9 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
39 ss2ixp 8950 . . . . . . . . 9 (∀𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) → X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)))
4038, 39ax-mp 5 . . . . . . . 8 X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
4128, 28xpex 7773 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42 fvex 6919 . . . . . . . . . . . . 13 (Hom ‘𝐶) ∈ V
4342rnex 7932 . . . . . . . . . . . 12 ran (Hom ‘𝐶) ∈ V
4443uniex 7761 . . . . . . . . . . 11 ran (Hom ‘𝐶) ∈ V
45 fvex 6919 . . . . . . . . . . . . 13 (Hom ‘𝐷) ∈ V
4645rnex 7932 . . . . . . . . . . . 12 ran (Hom ‘𝐷) ∈ V
4746uniex 7761 . . . . . . . . . . 11 ran (Hom ‘𝐷) ∈ V
4844, 47xpex 7773 . . . . . . . . . 10 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
4948pwex 5380 . . . . . . . . 9 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5041, 49ixpconst 8947 . . . . . . . 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)))
5140, 50sseqtri 4032 . . . . . . 7 X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))
52 eqid 2737 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 eqid 2737 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
5423, 52, 53, 25funcixp 17912 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)))
5551, 54sselid 3981 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))
5630, 55opelxpd 5724 . . . . 5 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → ⟨𝑓, 𝑔⟩ ∈ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))))
5756ex 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))))))
5822, 57biimtrrid 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))))))
5921, 58relssdv 5798 . 2 (𝜑 → (𝐶 Func 𝐷) ⊆ (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))))
601, 19, 59wunss 10752 1 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  wss 3951  𝒫 cpw 4600  cop 4632   cuni 4907   class class class wbr 5143   × cxp 5683  ran crn 5686  Rel wrel 5690  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  Xcixp 8937  WUnicwun 10740  ndxcnx 17230  Basecbs 17247  Hom chom 17308   Func cfunc 17899
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-wun 10742  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-dec 12734  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-func 17903
This theorem is referenced by:  wunnat  18004  catcfuccl  18163
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