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Theorem catcxpccl 17162
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 2799 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2799 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
4 eqid 2799 . . . . 5 (Hom ‘𝑋) = (Hom ‘𝑋)
5 eqid 2799 . . . . 5 (Hom ‘𝑌) = (Hom ‘𝑌)
6 eqid 2799 . . . . 5 (comp‘𝑋) = (comp‘𝑋)
7 eqid 2799 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
8 catcxpccl.x . . . . 5 (𝜑𝑋𝐵)
9 catcxpccl.y . . . . 5 (𝜑𝑌𝐵)
10 eqidd 2800 . . . . 5 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
111, 2, 3xpcbas 17133 . . . . . . 7 ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12 eqid 2799 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
131, 11, 4, 5, 12xpchomfval 17134 . . . . . 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 2800 . . . . 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 17132 . . . 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 16190 . . . . . . 7 Base = Slot 1
19 catcxpccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2017, 19wunndx 16205 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2118, 17, 20wunstr 16208 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
22 inss1 4028 . . . . . . . . 9 (𝑈 ∩ Cat) ⊆ 𝑈
23 catcxpccl.c . . . . . . . . . . 11 𝐶 = (CatCat‘𝑈)
24 catcxpccl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
2523, 24, 17catcbas 17061 . . . . . . . . . 10 (𝜑𝐵 = (𝑈 ∩ Cat))
268, 25eleqtrd 2880 . . . . . . . . 9 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
2722, 26sseldi 3796 . . . . . . . 8 (𝜑𝑋𝑈)
2818, 17, 27wunstr 16208 . . . . . . 7 (𝜑 → (Base‘𝑋) ∈ 𝑈)
299, 25eleqtrd 2880 . . . . . . . . 9 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
3022, 29sseldi 3796 . . . . . . . 8 (𝜑𝑌𝑈)
3118, 17, 30wunstr 16208 . . . . . . 7 (𝜑 → (Base‘𝑌) ∈ 𝑈)
3217, 28, 31wunxp 9834 . . . . . 6 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
3317, 21, 32wunop 9832 . . . . 5 (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
34 df-hom 16291 . . . . . . 7 Hom = Slot 14
3534, 17, 20wunstr 16208 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3617, 32, 32wunxp 9834 . . . . . . . 8 (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
3734, 17, 27wunstr 16208 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
3817, 37wunrn 9839 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
3917, 38wununi 9816 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑋) ∈ 𝑈)
4034, 17, 30wunstr 16208 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
4117, 40wunrn 9839 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
4217, 41wununi 9816 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑌) ∈ 𝑈)
4317, 39, 42wunxp 9834 . . . . . . . . 9 (𝜑 → ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
4417, 43wunpw 9817 . . . . . . . 8 (𝜑 → 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
45 ovssunirn 6913 . . . . . . . . . . . . 13 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ⊆ ran (Hom ‘𝑋)
46 ovssunirn 6913 . . . . . . . . . . . . 13 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ⊆ ran (Hom ‘𝑌)
47 xpss12 5327 . . . . . . . . . . . . 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)))
4845, 46, 47mp2an 684 . . . . . . . . . . . 12 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ⊆ ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
49 ovex 6910 . . . . . . . . . . . . . 14 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ∈ V
50 ovex 6910 . . . . . . . . . . . . . 14 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ∈ V
5149, 50xpex 7196 . . . . . . . . . . . . 13 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ V
5251elpw 4355 . . . . . . . . . . . 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)))
5348, 52mpbir 223 . . . . . . . . . . 11 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
5453rgen2w 3106 . . . . . . . . . 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))
55 eqid 2799 . . . . . . . . . . 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))))
5655fmpt2 7473 . . . . . . . . . 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)))
5754, 56mpbi 222 . . . . . . . . 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))
5857a1i 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)))
5917, 36, 44, 58wunf 9837 . . . . . . 7 (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))) ∈ 𝑈)
6013, 59syl5eqel 2882 . . . . . 6 (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
6117, 35, 60wunop 9832 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
62 df-cco 16292 . . . . . . 7 comp = Slot 15
6362, 17, 20wunstr 16208 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
6417, 36, 32wunxp 9834 . . . . . . 7 (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
6562, 17, 27wunstr 16208 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
6617, 65wunrn 9839 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
6717, 66wununi 9816 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
6817, 67wunrn 9839 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
6917, 68wununi 9816 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑋) ∈ 𝑈)
7017, 69wunpw 9817 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑋) ∈ 𝑈)
7162, 17, 30wunstr 16208 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
7217, 71wunrn 9839 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
7317, 72wununi 9816 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
7417, 73wunrn 9839 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
7517, 74wununi 9816 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
7617, 75wunpw 9817 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
7717, 70, 76wunxp 9834 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ 𝑈)
7817, 60wunrn 9839 . . . . . . . . . 10 (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
7917, 78wununi 9816 . . . . . . . . 9 (𝜑 ran (Hom ‘𝑇) ∈ 𝑈)
8017, 79, 79wunxp 9834 . . . . . . . 8 (𝜑 → ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ 𝑈)
8117, 77, 80wunpm 9835 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))) ∈ 𝑈)
82 fvex 6424 . . . . . . . . . . . . . . . . 17 (comp‘𝑋) ∈ V
8382rnex 7335 . . . . . . . . . . . . . . . 16 ran (comp‘𝑋) ∈ V
8483uniex 7187 . . . . . . . . . . . . . . 15 ran (comp‘𝑋) ∈ V
8584rnex 7335 . . . . . . . . . . . . . 14 ran ran (comp‘𝑋) ∈ V
8685uniex 7187 . . . . . . . . . . . . 13 ran ran (comp‘𝑋) ∈ V
8786pwex 5050 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑋) ∈ V
88 fvex 6424 . . . . . . . . . . . . . . . . 17 (comp‘𝑌) ∈ V
8988rnex 7335 . . . . . . . . . . . . . . . 16 ran (comp‘𝑌) ∈ V
9089uniex 7187 . . . . . . . . . . . . . . 15 ran (comp‘𝑌) ∈ V
9190rnex 7335 . . . . . . . . . . . . . 14 ran ran (comp‘𝑌) ∈ V
9291uniex 7187 . . . . . . . . . . . . 13 ran ran (comp‘𝑌) ∈ V
9392pwex 5050 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑌) ∈ V
9487, 93xpex 7196 . . . . . . . . . . 11 (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ V
95 fvex 6424 . . . . . . . . . . . . . 14 (Hom ‘𝑇) ∈ V
9695rnex 7335 . . . . . . . . . . . . 13 ran (Hom ‘𝑇) ∈ V
9796uniex 7187 . . . . . . . . . . . 12 ran (Hom ‘𝑇) ∈ V
9897, 97xpex 7196 . . . . . . . . . . 11 ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ V
99 ovssunirn 6913 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))
100 ovssunirn 6913 . . . . . . . . . . . . . . . . 17 (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋)
101 rnss 5557 . . . . . . . . . . . . . . . . 17 ((⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋) → ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋))
102 uniss 4651 . . . . . . . . . . . . . . . . 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))
103100, 101, 102mp2b 10 . . . . . . . . . . . . . . . 16 ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran ran (comp‘𝑋)
10499, 103sstri 3807 . . . . . . . . . . . . . . 15 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran ran (comp‘𝑋)
105 ovex 6910 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ V
106105elpw 4355 . . . . . . . . . . . . . . 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))
107104, 106mpbir 223 . . . . . . . . . . . . . 14 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋)
108 ovssunirn 6913 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))
109 ovssunirn 6913 . . . . . . . . . . . . . . . . 17 (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌)
110 rnss 5557 . . . . . . . . . . . . . . . . 17 ((⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌) → ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌))
111 uniss 4651 . . . . . . . . . . . . . . . . 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))
112109, 110, 111mp2b 10 . . . . . . . . . . . . . . . 16 ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran ran (comp‘𝑌)
113108, 112sstri 3807 . . . . . . . . . . . . . . 15 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran ran (comp‘𝑌)
114 ovex 6910 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ V
115114elpw 4355 . . . . . . . . . . . . . . 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))
116113, 115mpbir 223 . . . . . . . . . . . . . 14 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌)
117 opelxpi 5349 . . . . . . . . . . . . . 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)))
118107, 116, 117mp2an 684 . . . . . . . . . . . . 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))
119118rgen2w 3106 . . . . . . . . . . . 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))
120 eqid 2799 . . . . . . . . . . . . 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))⟩)
121120fmpt2 7473 . . . . . . . . . . . 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)))
122119, 121mpbi 222 . . . . . . . . . . 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))
123 ovssunirn 6913 . . . . . . . . . . . 12 ((2nd𝑥)(Hom ‘𝑇)𝑦) ⊆ ran (Hom ‘𝑇)
124 fvssunirn 6440 . . . . . . . . . . . 12 ((Hom ‘𝑇)‘𝑥) ⊆ ran (Hom ‘𝑇)
125 xpss12 5327 . . . . . . . . . . . 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)))
126123, 124, 125mp2an 684 . . . . . . . . . . 11 (((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))
127 elpm2r 8113 . . . . . . . . . . 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))))
12894, 98, 122, 126, 127mp4an 685 . . . . . . . . . 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)))
129128rgen2w 3106 . . . . . . . . 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)))
130 eqid 2799 . . . . . . . . . 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))⟩))
131130fmpt2 7473 . . . . . . . . 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))))
132129, 131mpbi 222 . . . . . . . 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)))
133132a1i 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))))
13417, 64, 81, 133wunf 9837 . . . . . 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)
13517, 63, 134wunop 9832 . . . . 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)
13617, 33, 61, 135wuntp 9821 . . . 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)
13716, 136eqeltrd 2878 . . 3 (𝜑𝑇𝑈)
138 inss2 4029 . . . . 5 (𝑈 ∩ Cat) ⊆ Cat
139138, 26sseldi 3796 . . . 4 (𝜑𝑋 ∈ Cat)
140138, 29sseldi 3796 . . . 4 (𝜑𝑌 ∈ Cat)
1411, 139, 140xpccat 17145 . . 3 (𝜑𝑇 ∈ Cat)
142137, 141elind 3996 . 2 (𝜑𝑇 ∈ (𝑈 ∩ Cat))
143142, 25eleqtrrd 2881 1 (𝜑𝑇𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  cin 3768  wss 3769  𝒫 cpw 4349  {ctp 4372  cop 4374   cuni 4628   × cxp 5310  ran crn 5313  wf 6097  cfv 6101  (class class class)co 6878  cmpt2 6880  ωcom 7299  1st c1st 7399  2nd c2nd 7400  pm cpm 8096  WUnicwun 9810  1c1 10225  4c4 11370  5c5 11371  cdc 11783  ndxcnx 16181  Basecbs 16184  Hom chom 16278  compcco 16279  Catccat 16639  CatCatccatc 17058   ×c cxpc 17123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-omul 7804  df-er 7982  df-ec 7984  df-qs 7988  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-wun 9812  df-ni 9982  df-pli 9983  df-mi 9984  df-lti 9985  df-plpq 10018  df-mpq 10019  df-ltpq 10020  df-enq 10021  df-nq 10022  df-erq 10023  df-plq 10024  df-mq 10025  df-1nq 10026  df-rq 10027  df-ltnq 10028  df-np 10091  df-plp 10093  df-ltp 10095  df-enr 10165  df-nr 10166  df-c 10230  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-dec 11784  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-hom 16291  df-cco 16292  df-cat 16643  df-cid 16644  df-catc 17059  df-xpc 17127
This theorem is referenced by: (None)
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