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Theorem wunfuncOLD 17643
Description: Obsolete proof of wunfunc 17642 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
wunfunc.1 (𝜑𝑈 ∈ WUni)
wunfunc.2 (𝜑𝐶𝑈)
wunfunc.3 (𝜑𝐷𝑈)
Assertion
Ref Expression
wunfuncOLD (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)

Proof of Theorem wunfuncOLD
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 16941 . . . . 5 Base = Slot 1
3 wunfunc.3 . . . . 5 (𝜑𝐷𝑈)
42, 1, 3wunstr 16917 . . . 4 (𝜑 → (Base‘𝐷) ∈ 𝑈)
5 wunfunc.2 . . . . 5 (𝜑𝐶𝑈)
62, 1, 5wunstr 16917 . . . 4 (𝜑 → (Base‘𝐶) ∈ 𝑈)
71, 4, 6wunmap 10510 . . 3 (𝜑 → ((Base‘𝐷) ↑m (Base‘𝐶)) ∈ 𝑈)
8 df-hom 17014 . . . . . . . . 9 Hom = Slot 14
98, 1, 5wunstr 16917 . . . . . . . 8 (𝜑 → (Hom ‘𝐶) ∈ 𝑈)
101, 9wunrn 10513 . . . . . . 7 (𝜑 → ran (Hom ‘𝐶) ∈ 𝑈)
111, 10wununi 10490 . . . . . 6 (𝜑 ran (Hom ‘𝐶) ∈ 𝑈)
128, 1, 3wunstr 16917 . . . . . . . 8 (𝜑 → (Hom ‘𝐷) ∈ 𝑈)
131, 12wunrn 10513 . . . . . . 7 (𝜑 → ran (Hom ‘𝐷) ∈ 𝑈)
141, 13wununi 10490 . . . . . 6 (𝜑 ran (Hom ‘𝐷) ∈ 𝑈)
151, 11, 14wunxp 10508 . . . . 5 (𝜑 → ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
161, 15wunpw 10491 . . . 4 (𝜑 → 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ 𝑈)
171, 6, 6wunxp 10508 . . . 4 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) ∈ 𝑈)
181, 16, 17wunmap 10510 . . 3 (𝜑 → (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))) ∈ 𝑈)
191, 7, 18wunxp 10508 . 2 (𝜑 → (((Base‘𝐷) ↑m (Base‘𝐶)) × (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶)))) ∈ 𝑈)
20 relfunc 17605 . . . 4 Rel (𝐶 Func 𝐷)
2120a1i 11 . . 3 (𝜑 → Rel (𝐶 Func 𝐷))
22 df-br 5078 . . . 4 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
23 eqid 2733 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
24 eqid 2733 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
25 simpr 484 . . . . . . . 8 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
2623, 24, 25funcf1 17609 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
27 fvex 6805 . . . . . . . 8 (Base‘𝐷) ∈ V
28 fvex 6805 . . . . . . . 8 (Base‘𝐶) ∈ V
2927, 28elmap 8679 . . . . . . 7 (𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)) ↔ 𝑓:(Base‘𝐶)⟶(Base‘𝐷))
3026, 29sylibr 233 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑓 ∈ ((Base‘𝐷) ↑m (Base‘𝐶)))
31 mapsspw 8686 . . . . . . . . . . 11 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))))
32 fvssunirn 6823 . . . . . . . . . . . . 13 ((Hom ‘𝐶)‘𝑧) ⊆ ran (Hom ‘𝐶)
33 ovssunirn 7331 . . . . . . . . . . . . 13 ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ⊆ ran (Hom ‘𝐷)
34 xpss12 5606 . . . . . . . . . . . . 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 688 . . . . . . . . . . . 12 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3635sspwi 4550 . . . . . . . . . . 11 𝒫 (((Hom ‘𝐶)‘𝑧) × ((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧)))) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3731, 36sstri 3932 . . . . . . . . . 10 (((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)) ⊆ 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷))
3837rgenw 3063 . . . . . . . . 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 8718 . . . . . . . . 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 7623 . . . . . . . . 9 ((Base‘𝐶) × (Base‘𝐶)) ∈ V
42 fvex 6805 . . . . . . . . . . . . 13 (Hom ‘𝐶) ∈ V
4342rnex 7779 . . . . . . . . . . . 12 ran (Hom ‘𝐶) ∈ V
4443uniex 7614 . . . . . . . . . . 11 ran (Hom ‘𝐶) ∈ V
45 fvex 6805 . . . . . . . . . . . . 13 (Hom ‘𝐷) ∈ V
4645rnex 7779 . . . . . . . . . . . 12 ran (Hom ‘𝐷) ∈ V
4746uniex 7614 . . . . . . . . . . 11 ran (Hom ‘𝐷) ∈ V
4844, 47xpex 7623 . . . . . . . . . 10 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
4948pwex 5306 . . . . . . . . 9 𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ∈ V
5041, 49ixpconst 8715 . . . . . . . 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 3959 . . . . . . 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 2733 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
53 eqid 2733 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
5423, 52, 53, 25funcixp 17610 . . . . . . 7 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔X𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))(((𝑓‘(1st𝑧))(Hom ‘𝐷)(𝑓‘(2nd𝑧))) ↑m ((Hom ‘𝐶)‘𝑧)))
5551, 54sselid 3921 . . . . . 6 ((𝜑𝑓(𝐶 Func 𝐷)𝑔) → 𝑔 ∈ (𝒫 ( ran (Hom ‘𝐶) × ran (Hom ‘𝐷)) ↑m ((Base‘𝐶) × (Base‘𝐶))))
5630, 55opelxpd 5629 . . . . 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, 57syl5bir 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))))))
5921, 58relssdv 5701 . 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 10496 1 (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2101  wral 3059  wss 3889  𝒫 cpw 4536  cop 4570   cuni 4841   class class class wbr 5077   × cxp 5589  ran crn 5592  Rel wrel 5596  wf 6443  cfv 6447  (class class class)co 7295  1st c1st 7849  2nd c2nd 7850  m cmap 8635  Xcixp 8705  WUnicwun 10484  1c1 10900  4c4 12058  cdc 12465  Basecbs 16940  Hom chom 17001   Func cfunc 17597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-map 8637  df-pm 8638  df-ixp 8706  df-wun 10486  df-slot 16911  df-base 16941  df-hom 17014  df-func 17601
This theorem is referenced by: (None)
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