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Theorem wunfunc 16760
Description: A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
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 df-base 16065 . . . . 5 Base = Slot 1
3 wunfunc.3 . . . . 5 (𝜑𝐷𝑈)
42, 1, 3wunstr 16083 . . . 4 (𝜑 → (Base‘𝐷) ∈ 𝑈)
5 wunfunc.2 . . . . 5 (𝜑𝐶𝑈)
62, 1, 5wunstr 16083 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
71, 4, 6wunmap 9740 . . 3 (𝜑 → ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
8 df-hom 16168 . . . . . . . . 9 Hom = Slot 14
98, 1, 5wunstr 16083 . . . . . . . 8 (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
101, 9wunrn 9743 . . . . . . 7 (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
111, 10wununi 9720 . . . . . 6 (𝜑 ran (Hom ‘𝐶) ∈ 𝑈)
128, 1, 3wunstr 16083 . . . . . . . 8 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
131, 12wunrn 9743 . . . . . . 7 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
141, 13wununi 9720 . . . . . 6 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
151, 11, 14wunxp 9738 . . . . 5 (𝜑 → ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
161, 15wunpw 9721 . . . 4 (𝜑 → 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
171, 6, 6wunxp 9738 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
181, 16, 17wunmap 9740 . . 3 (𝜑 → (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
191, 7, 18wunxp 9738 . 2 (𝜑 → (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20 relfunc 16723 . . . 4 Rel (𝐶 Func 𝐷)
2120a1i 11 . . 3 (𝜑 → Rel (𝐶 Func 𝐷))
22 df-br 4805 . . . 4 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23 eqid 2760 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
24 eqid 2760 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
25 simpr 479 . . . . . . . 8 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
2623, 24, 25funcf1 16727 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27 fvex 6362 . . . . . . . 8 (Base‘𝐷) ∈ V
28 fvex 6362 . . . . . . . 8 (Base‘𝐶) ∈ V
2927, 28elmap 8052 . . . . . . 7 (𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
3026, 29sylibr 224 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)))
31 mapsspw 8059 . . . . . . . . . . 11 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))))
32 fvssunirn 6378 . . . . . . . . . . . . 13 ((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶)
33 ovssunirn 6844 . . . . . . . . . . . . 13 ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)
34 xpss12 5281 . . . . . . . . . . . . 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 710 . . . . . . . . . . . 12 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
36 sspwb 5066 . . . . . . . . . . . 12 ((((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↔ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)))
3735, 36mpbi 220 . . . . . . . . . . 11 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3831, 37sstri 3753 . . . . . . . . . 10 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3938rgenw 3062 . . . . . . . . 9 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
40 ss2ixp 8087 . . . . . . . . 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)))
4139, 40ax-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))
4228, 28xpex 7127 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
43 fvex 6362 . . . . . . . . . . . . 13 (Hom ‘𝐶) ∈ V
4443rnex 7265 . . . . . . . . . . . 12 ran (Hom ‘𝐶) ∈ V
4544uniex 7118 . . . . . . . . . . 11 ran (Hom ‘𝐶) ∈ V
46 fvex 6362 . . . . . . . . . . . . 13 (Hom ‘𝐷) ∈ V
4746rnex 7265 . . . . . . . . . . . 12 ran (Hom ‘𝐷) ∈ V
4847uniex 7118 . . . . . . . . . . 11 ran (Hom ‘𝐷) ∈ V
4945, 48xpex 7127 . . . . . . . . . 10 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5049pwex 4997 . . . . . . . . 9 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5142, 50ixpconst 8084 . . . . . . . 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)))
5241, 51sseqtri 3778 . . . . . . 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)))
53 eqid 2760 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
54 eqid 2760 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
5523, 53, 54, 25funcixp 16728 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
5652, 55sseldi 3742 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))))
57 opelxpi 5305 . . . . . 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)))))
5830, 56, 57syl2anc 696 . . . . 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)))))
5958ex 449 . . . 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))))))
6022, 59syl5bir 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))))))
6121, 60relssdv 5369 . 2 (𝜑 → (𝐶 Func 𝐷) ⊆ (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))))
621, 19, 61wunss 9726 1 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wral 3050  wss 3715  𝒫 cpw 4302  cop 4327   cuni 4588   class class class wbr 4804   × cxp 5264  ran crn 5267  Rel wrel 5271  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  Xcixp 8074  WUnicwun 9714  1c1 10129  4c4 11264  cdc 11685  Basecbs 16059  Hom chom 16154   Func cfunc 16715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-pm 8026  df-ixp 8075  df-wun 9716  df-slot 16063  df-base 16065  df-hom 16168  df-func 16719
This theorem is referenced by:  wunnat  16817  catcfuccl  16960
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