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Theorem catcxpccl 17436
Description: The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcxpccl.c 𝐶 = (CatCat‘𝑈)
catcxpccl.b 𝐵 = (Base‘𝐶)
catcxpccl.o 𝑇 = (𝑋 ×c 𝑌)
catcxpccl.u (𝜑𝑈 ∈ WUni)
catcxpccl.1 (𝜑 → ω ∈ 𝑈)
catcxpccl.x (𝜑𝑋𝐵)
catcxpccl.y (𝜑𝑌𝐵)
Assertion
Ref Expression
catcxpccl (𝜑𝑇𝐵)

Proof of Theorem catcxpccl
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcxpccl.o . . . . 5 𝑇 = (𝑋 ×c 𝑌)
2 eqid 2820 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2820 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
4 eqid 2820 . . . . 5 (Hom ‘𝑋) = (Hom ‘𝑋)
5 eqid 2820 . . . . 5 (Hom ‘𝑌) = (Hom ‘𝑌)
6 eqid 2820 . . . . 5 (comp‘𝑋) = (comp‘𝑋)
7 eqid 2820 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
8 catcxpccl.x . . . . 5 (𝜑𝑋𝐵)
9 catcxpccl.y . . . . 5 (𝜑𝑌𝐵)
10 eqidd 2821 . . . . 5 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
111, 2, 3xpcbas 17407 . . . . . . 7 ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12 eqid 2820 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
131, 11, 4, 5, 12xpchomfval 17408 . . . . . 6 (Hom ‘𝑇) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))))
1413a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑇) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))))
15 eqidd 2821 . . . . 5 (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 17406 . . . 4 (𝜑𝑇 = {⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩, ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩))⟩})
17 catcxpccl.u . . . . 5 (𝜑𝑈 ∈ WUni)
18 df-base 16468 . . . . . . 7 Base = Slot 1
19 catcxpccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2017, 19wunndx 16483 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2118, 17, 20wunstr 16486 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
22 catcxpccl.c . . . . . . . . . . 11 𝐶 = (CatCat‘𝑈)
23 catcxpccl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
2422, 23, 17catcbas 17336 . . . . . . . . . 10 (𝜑𝐵 = (𝑈 ∩ Cat))
258, 24eleqtrd 2913 . . . . . . . . 9 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
2625elin1d 4153 . . . . . . . 8 (𝜑𝑋𝑈)
2718, 17, 26wunstr 16486 . . . . . . 7 (𝜑 → (Base‘𝑋) ∈ 𝑈)
289, 24eleqtrd 2913 . . . . . . . . 9 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
2928elin1d 4153 . . . . . . . 8 (𝜑𝑌𝑈)
3018, 17, 29wunstr 16486 . . . . . . 7 (𝜑 → (Base‘𝑌) ∈ 𝑈)
3117, 27, 30wunxp 10124 . . . . . 6 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
3217, 21, 31wunop 10122 . . . . 5 (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
33 df-hom 16568 . . . . . . 7 Hom = Slot 14
3433, 17, 20wunstr 16486 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3517, 31, 31wunxp 10124 . . . . . . . 8 (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
3633, 17, 26wunstr 16486 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
3717, 36wunrn 10129 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
3817, 37wununi 10106 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑋) ∈ 𝑈)
3933, 17, 29wunstr 16486 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
4017, 39wunrn 10129 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
4117, 40wununi 10106 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑌) ∈ 𝑈)
4217, 38, 41wunxp 10124 . . . . . . . . 9 (𝜑 → ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
4317, 42wunpw 10107 . . . . . . . 8 (𝜑 → 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
44 ovssunirn 7169 . . . . . . . . . . . . 13 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ⊆ ran (Hom ‘𝑋)
45 ovssunirn 7169 . . . . . . . . . . . . 13 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ⊆ ran (Hom ‘𝑌)
46 xpss12 5546 . . . . . . . . . . . . 13 ((((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ⊆ ran (Hom ‘𝑋) ∧ ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ⊆ ran (Hom ‘𝑌)) → (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ⊆ ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)))
4744, 45, 46mp2an 690 . . . . . . . . . . . 12 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ⊆ ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
48 ovex 7166 . . . . . . . . . . . . . 14 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ∈ V
49 ovex 7166 . . . . . . . . . . . . . 14 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ∈ V
5048, 49xpex 7454 . . . . . . . . . . . . 13 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ V
5150elpw 4519 . . . . . . . . . . . 12 ((((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ↔ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ⊆ ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)))
5247, 51mpbir 233 . . . . . . . . . . 11 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
5352rgen2w 3138 . . . . . . . . . 10 𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌))∀𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌))(((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
54 eqid 2820 . . . . . . . . . . 11 (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))))
5554fmpo 7744 . . . . . . . . . 10 (∀𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌))∀𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌))(((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ↔ (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)))
5653, 55mpbi 232 . . . . . . . . 9 (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
5756a1i 11 . . . . . . . 8 (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)))
5817, 35, 43, 57wunf 10127 . . . . . . 7 (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))) ∈ 𝑈)
5913, 58eqeltrid 2915 . . . . . 6 (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
6017, 34, 59wunop 10122 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
61 df-cco 16569 . . . . . . 7 comp = Slot 15
6261, 17, 20wunstr 16486 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
6317, 35, 31wunxp 10124 . . . . . . 7 (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
6461, 17, 26wunstr 16486 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
6517, 64wunrn 10129 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
6617, 65wununi 10106 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
6717, 66wunrn 10129 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
6817, 67wununi 10106 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑋) ∈ 𝑈)
6917, 68wunpw 10107 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑋) ∈ 𝑈)
7061, 17, 29wunstr 16486 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
7117, 70wunrn 10129 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
7217, 71wununi 10106 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
7317, 72wunrn 10129 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
7417, 73wununi 10106 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
7517, 74wunpw 10107 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
7617, 69, 75wunxp 10124 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ 𝑈)
7717, 59wunrn 10129 . . . . . . . . . 10 (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
7817, 77wununi 10106 . . . . . . . . 9 (𝜑 ran (Hom ‘𝑇) ∈ 𝑈)
7917, 78, 78wunxp 10124 . . . . . . . 8 (𝜑 → ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ 𝑈)
8017, 76, 79wunpm 10125 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))) ∈ 𝑈)
81 fvex 6659 . . . . . . . . . . . . . . . . 17 (comp‘𝑋) ∈ V
8281rnex 7595 . . . . . . . . . . . . . . . 16 ran (comp‘𝑋) ∈ V
8382uniex 7445 . . . . . . . . . . . . . . 15 ran (comp‘𝑋) ∈ V
8483rnex 7595 . . . . . . . . . . . . . 14 ran ran (comp‘𝑋) ∈ V
8584uniex 7445 . . . . . . . . . . . . 13 ran ran (comp‘𝑋) ∈ V
8685pwex 5257 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑋) ∈ V
87 fvex 6659 . . . . . . . . . . . . . . . . 17 (comp‘𝑌) ∈ V
8887rnex 7595 . . . . . . . . . . . . . . . 16 ran (comp‘𝑌) ∈ V
8988uniex 7445 . . . . . . . . . . . . . . 15 ran (comp‘𝑌) ∈ V
9089rnex 7595 . . . . . . . . . . . . . 14 ran ran (comp‘𝑌) ∈ V
9190uniex 7445 . . . . . . . . . . . . 13 ran ran (comp‘𝑌) ∈ V
9291pwex 5257 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑌) ∈ V
9386, 92xpex 7454 . . . . . . . . . . 11 (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ V
94 fvex 6659 . . . . . . . . . . . . . 14 (Hom ‘𝑇) ∈ V
9594rnex 7595 . . . . . . . . . . . . 13 ran (Hom ‘𝑇) ∈ V
9695uniex 7445 . . . . . . . . . . . 12 ran (Hom ‘𝑇) ∈ V
9796, 96xpex 7454 . . . . . . . . . . 11 ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ V
98 ovssunirn 7169 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))
99 ovssunirn 7169 . . . . . . . . . . . . . . . . 17 (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋)
100 rnss 5785 . . . . . . . . . . . . . . . . 17 ((⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋) → ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋))
101 uniss 4822 . . . . . . . . . . . . . . . . 17 (ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋) → ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋))
10299, 100, 101mp2b 10 . . . . . . . . . . . . . . . 16 ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋)
10398, 102sstri 3955 . . . . . . . . . . . . . . 15 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran ran (comp‘𝑋)
104 ovex 7166 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ V
105104elpw 4519 . . . . . . . . . . . . . . 15 (((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋) ↔ ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran ran (comp‘𝑋))
106103, 105mpbir 233 . . . . . . . . . . . . . 14 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋)
107 ovssunirn 7169 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))
108 ovssunirn 7169 . . . . . . . . . . . . . . . . 17 (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌)
109 rnss 5785 . . . . . . . . . . . . . . . . 17 ((⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌) → ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌))
110 uniss 4822 . . . . . . . . . . . . . . . . 17 (ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌) → ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌))
111108, 109, 110mp2b 10 . . . . . . . . . . . . . . . 16 ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌)
112107, 111sstri 3955 . . . . . . . . . . . . . . 15 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran ran (comp‘𝑌)
113 ovex 7166 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ V
114113elpw 4519 . . . . . . . . . . . . . . 15 (((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌) ↔ ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran ran (comp‘𝑌))
115112, 114mpbir 233 . . . . . . . . . . . . . 14 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌)
116 opelxpi 5568 . . . . . . . . . . . . . 14 ((((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋) ∧ ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌)) → ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩ ∈ (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)))
117106, 115, 116mp2an 690 . . . . . . . . . . . . 13 ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩ ∈ (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌))
118117rgen2w 3138 . . . . . . . . . . . 12 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦)∀𝑓 ∈ ((Hom ‘𝑇)‘𝑥)⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩ ∈ (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌))
119 eqid 2820 . . . . . . . . . . . . 13 (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩) = (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)
120119fmpo 7744 . . . . . . . . . . . 12 (∀𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦)∀𝑓 ∈ ((Hom ‘𝑇)‘𝑥)⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩ ∈ (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↔ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩):(((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)))
121118, 120mpbi 232 . . . . . . . . . . 11 (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩):(((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌))
122 ovssunirn 7169 . . . . . . . . . . . 12 ((2nd𝑥)(Hom ‘𝑇)𝑦) ⊆ ran (Hom ‘𝑇)
123 fvssunirn 6675 . . . . . . . . . . . 12 ((Hom ‘𝑇)‘𝑥) ⊆ ran (Hom ‘𝑇)
124 xpss12 5546 . . . . . . . . . . . 12 ((((2nd𝑥)(Hom ‘𝑇)𝑦) ⊆ ran (Hom ‘𝑇) ∧ ((Hom ‘𝑇)‘𝑥) ⊆ ran (Hom ‘𝑇)) → (((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)))
125122, 123, 124mp2an 690 . . . . . . . . . . 11 (((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))
126 elpm2r 8402 . . . . . . . . . . 11 ((((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ V ∧ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ V) ∧ ((𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩):(((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∧ (((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)))) → (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩) ∈ ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))))
12793, 97, 121, 125, 126mp4an 691 . . . . . . . . . 10 (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩) ∈ ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)))
128127rgen2w 3138 . . . . . . . . 9 𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))∀𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌))(𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩) ∈ ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)))
129 eqid 2820 . . . . . . . . . 10 (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)) = (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩))
130129fmpo 7744 . . . . . . . . 9 (∀𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))∀𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌))(𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩) ∈ ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))) ↔ (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))))
131128, 130mpbi 232 . . . . . . . 8 (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)))
132131a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))))
13317, 63, 80, 132wunf 10127 . . . . . 6 (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩)) ∈ 𝑈)
13417, 62, 133wunop 10122 . . . . 5 (𝜑 → ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩))⟩ ∈ 𝑈)
13517, 32, 60, 134wuntp 10111 . . . 4 (𝜑 → {⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩, ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓))⟩))⟩} ∈ 𝑈)
13616, 135eqeltrd 2911 . . 3 (𝜑𝑇𝑈)
13725elin2d 4154 . . . 4 (𝜑𝑋 ∈ Cat)
13828elin2d 4154 . . . 4 (𝜑𝑌 ∈ Cat)
1391, 137, 138xpccat 17419 . . 3 (𝜑𝑇 ∈ Cat)
140136, 139elind 4149 . 2 (𝜑𝑇 ∈ (𝑈 ∩ Cat))
141140, 24eleqtrrd 2914 1 (𝜑𝑇𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  cin 3912  wss 3913  𝒫 cpw 4515  {ctp 4547  cop 4549   cuni 4814   × cxp 5529  ran crn 5532  wf 6327  cfv 6331  (class class class)co 7133  cmpo 7135  ωcom 7558  1st c1st 7665  2nd c2nd 7666  pm cpm 8385  WUnicwun 10100  1c1 10516  4c4 11673  5c5 11674  cdc 12077  ndxcnx 16459  Basecbs 16462  Hom chom 16555  compcco 16556  Catccat 16914  CatCatccatc 17333   ×c cxpc 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-omul 8085  df-er 8267  df-ec 8269  df-qs 8273  df-map 8386  df-pm 8387  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-wun 10102  df-ni 10272  df-pli 10273  df-mi 10274  df-lti 10275  df-plpq 10308  df-mpq 10309  df-ltpq 10310  df-enq 10311  df-nq 10312  df-erq 10313  df-plq 10314  df-mq 10315  df-1nq 10316  df-rq 10317  df-ltnq 10318  df-np 10381  df-plp 10383  df-ltp 10385  df-enr 10455  df-nr 10456  df-c 10521  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-fz 12877  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-hom 16568  df-cco 16569  df-cat 16918  df-cid 16919  df-catc 17334  df-xpc 17401
This theorem is referenced by: (None)
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