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Theorem wunfunc 16382
 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 15700 . . . . 5 Base = Slot 1
3 wunfunc.3 . . . . 5 (𝜑𝐷𝑈)
42, 1, 3wunstr 15714 . . . 4 (𝜑 → (Base‘𝐷) ∈ 𝑈)
5 wunfunc.2 . . . . 5 (𝜑𝐶𝑈)
62, 1, 5wunstr 15714 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
71, 4, 6wunmap 9427 . . 3 (𝜑 → ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ∈ 𝑈)
8 df-hom 15793 . . . . . . . . 9 Hom = Slot 14
98, 1, 5wunstr 15714 . . . . . . . 8 (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
101, 9wunrn 9430 . . . . . . 7 (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
111, 10wununi 9407 . . . . . 6 (𝜑 ran (Hom ‘𝐶) ∈ 𝑈)
128, 1, 3wunstr 15714 . . . . . . . 8 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
131, 12wunrn 9430 . . . . . . 7 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
141, 13wununi 9407 . . . . . 6 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
151, 11, 14wunxp 9425 . . . . 5 (𝜑 → ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
161, 15wunpw 9408 . . . 4 (𝜑 → 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
171, 6, 6wunxp 9425 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
181, 16, 17wunmap 9427 . . 3 (𝜑 → (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
191, 7, 18wunxp 9425 . 2 (𝜑 → (((Base‘𝐷) ↑𝑚 (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20 relfunc 16345 . . . 4 Rel (𝐶 Func 𝐷)
2120a1i 11 . . 3 (𝜑 → Rel (𝐶 Func 𝐷))
22 df-br 4584 . . . 4 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23 eqid 2610 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
24 eqid 2610 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
25 simpr 476 . . . . . . . 8 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
2623, 24, 25funcf1 16349 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27 fvex 6113 . . . . . . . 8 (Base‘𝐷) ∈ V
28 fvex 6113 . . . . . . . 8 (Base‘𝐶) ∈ V
2927, 28elmap 7772 . . . . . . 7 (𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
3026, 29sylibr 223 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑𝑚 (Base‘𝐶)))
31 mapsspw 7779 . . . . . . . . . . 11 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))))
32 fvssunirn 6127 . . . . . . . . . . . . 13 ((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶)
33 ovssunirn 6579 . . . . . . . . . . . . 13 ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)
34 xpss12 5148 . . . . . . . . . . . . 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 704 . . . . . . . . . . . 12 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
36 sspwb 4844 . . . . . . . . . . . 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 219 . . . . . . . . . . 11 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3831, 37sstri 3577 . . . . . . . . . 10 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3938rgenw 2908 . . . . . . . . 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 7807 . . . . . . . . 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 6860 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
43 fvex 6113 . . . . . . . . . . . . 13 (Hom ‘𝐶) ∈ V
4443rnex 6992 . . . . . . . . . . . 12 ran (Hom ‘𝐶) ∈ V
4544uniex 6851 . . . . . . . . . . 11 ran (Hom ‘𝐶) ∈ V
46 fvex 6113 . . . . . . . . . . . . 13 (Hom ‘𝐷) ∈ V
4746rnex 6992 . . . . . . . . . . . 12 ran (Hom ‘𝐷) ∈ V
4847uniex 6851 . . . . . . . . . . 11 ran (Hom ‘𝐷) ∈ V
4945, 48xpex 6860 . . . . . . . . . 10 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5049pwex 4774 . . . . . . . . 9 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5142, 50ixpconst 7804 . . . . . . . 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 3600 . . . . . . 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 2610 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
54 eqid 2610 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
5523, 53, 54, 25funcixp 16350 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝐶)‘𝑧)))
5652, 55sseldi 3566 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑𝑚 ((Base‘𝐶) × (Base‘𝐶))))
57 opelxpi 5072 . . . . . 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 691 . . . . 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 232 . . 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 5135 . 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 9413 1 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039  Rel wrel 5043  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Xcixp 7794  WUnicwun 9401  1c1 9816  4c4 10949  ;cdc 11369  Basecbs 15695  Hom chom 15779   Func cfunc 16337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-pm 7747  df-ixp 7795  df-wun 9403  df-slot 15699  df-base 15700  df-hom 15793  df-func 16341 This theorem is referenced by:  wunnat  16439  catcfuccl  16582
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