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Theorem wunfunc 16606
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 15910 . . . . 5 Base = Slot 1
3 wunfunc.3 . . . . 5 (𝜑𝐷𝑈)
42, 1, 3wunstr 15928 . . . 4 (𝜑 → (Base‘𝐷) ∈ 𝑈)
5 wunfunc.2 . . . . 5 (𝜑𝐶𝑈)
62, 1, 5wunstr 15928 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
71, 4, 6wunmap 9586 . . 3 (𝜑 → ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
8 df-hom 16013 . . . . . . . . 9 Hom = Slot 14
98, 1, 5wunstr 15928 . . . . . . . 8 (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
101, 9wunrn 9589 . . . . . . 7 (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
111, 10wununi 9566 . . . . . 6 (𝜑 ran (Hom ‘𝐶) ∈ 𝑈)
128, 1, 3wunstr 15928 . . . . . . . 8 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
131, 12wunrn 9589 . . . . . . 7 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
141, 13wununi 9566 . . . . . 6 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
151, 11, 14wunxp 9584 . . . . 5 (𝜑 → ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
161, 15wunpw 9567 . . . 4 (𝜑 → 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
171, 6, 6wunxp 9584 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
181, 16, 17wunmap 9586 . . 3 (𝜑 → (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
191, 7, 18wunxp 9584 . 2 (𝜑 → (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20 relfunc 16569 . . . 4 Rel (𝐶 Func 𝐷)
2120a1i 11 . . 3 (𝜑 → Rel (𝐶 Func 𝐷))
22 df-br 4686 . . . 4 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23 eqid 2651 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
24 eqid 2651 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
25 simpr 476 . . . . . . . 8 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
2623, 24, 25funcf1 16573 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27 fvex 6239 . . . . . . . 8 (Base‘𝐷) ∈ V
28 fvex 6239 . . . . . . . 8 (Base‘𝐶) ∈ V
2927, 28elmap 7928 . . . . . . 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 7935 . . . . . . . . . . 11 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))))
32 fvssunirn 6255 . . . . . . . . . . . . 13 ((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶)
33 ovssunirn 6721 . . . . . . . . . . . . 13 ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)
34 xpss12 5158 . . . . . . . . . . . . 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 708 . . . . . . . . . . . 12 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
36 sspwb 4947 . . . . . . . . . . . 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 3645 . . . . . . . . . 10 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3938rgenw 2953 . . . . . . . . 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 7963 . . . . . . . . 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 7004 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
43 fvex 6239 . . . . . . . . . . . . 13 (Hom ‘𝐶) ∈ V
4443rnex 7142 . . . . . . . . . . . 12 ran (Hom ‘𝐶) ∈ V
4544uniex 6995 . . . . . . . . . . 11 ran (Hom ‘𝐶) ∈ V
46 fvex 6239 . . . . . . . . . . . . 13 (Hom ‘𝐷) ∈ V
4746rnex 7142 . . . . . . . . . . . 12 ran (Hom ‘𝐷) ∈ V
4847uniex 6995 . . . . . . . . . . 11 ran (Hom ‘𝐷) ∈ V
4945, 48xpex 7004 . . . . . . . . . 10 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5049pwex 4878 . . . . . . . . 9 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5142, 50ixpconst 7960 . . . . . . . 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 3670 . . . . . . 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 2651 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
54 eqid 2651 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
5523, 53, 54, 25funcixp 16574 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
5652, 55sseldi 3634 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))))
57 opelxpi 5182 . . . . . 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 694 . . . . 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 5246 . 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 9572 1 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  wss 3607  𝒫 cpw 4191  cop 4216   cuni 4468   class class class wbr 4685   × cxp 5141  ran crn 5144  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  WUnicwun 9560  1c1 9975  4c4 11110  cdc 11531  Basecbs 15904  Hom chom 15999   Func cfunc 16561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-pm 7902  df-ixp 7951  df-wun 9562  df-slot 15908  df-base 15910  df-hom 16013  df-func 16565
This theorem is referenced by:  wunnat  16663  catcfuccl  16806
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