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Theorem catcxpcclOLD 18193
Description: Obsolete proof of catcxpccl 18192 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
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
catcxpcclOLD (𝜑𝑇𝐵)

Proof of Theorem catcxpcclOLD
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 2725 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2725 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
4 eqid 2725 . . . . 5 (Hom ‘𝑋) = (Hom ‘𝑋)
5 eqid 2725 . . . . 5 (Hom ‘𝑌) = (Hom ‘𝑌)
6 eqid 2725 . . . . 5 (comp‘𝑋) = (comp‘𝑋)
7 eqid 2725 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
8 catcxpccl.x . . . . 5 (𝜑𝑋𝐵)
9 catcxpccl.y . . . . 5 (𝜑𝑌𝐵)
10 eqidd 2726 . . . . 5 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
111, 2, 3xpcbas 18163 . . . . . . 7 ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12 eqid 2725 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
131, 11, 4, 5, 12xpchomfval 18164 . . . . . 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 2726 . . . . 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 18162 . . . 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 17175 . . . . . . 7 Base = Slot 1
19 catcxpccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2017, 19wunndx 17158 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2118, 17, 20wunstr 17151 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
22 catcxpccl.c . . . . . . . . . . 11 𝐶 = (CatCat‘𝑈)
23 catcxpccl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
2422, 23, 17catcbas 18084 . . . . . . . . . 10 (𝜑𝐵 = (𝑈 ∩ Cat))
258, 24eleqtrd 2827 . . . . . . . . 9 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
2625elin1d 4193 . . . . . . . 8 (𝜑𝑋𝑈)
2718, 17, 26wunstr 17151 . . . . . . 7 (𝜑 → (Base‘𝑋) ∈ 𝑈)
289, 24eleqtrd 2827 . . . . . . . . 9 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
2928elin1d 4193 . . . . . . . 8 (𝜑𝑌𝑈)
3018, 17, 29wunstr 17151 . . . . . . 7 (𝜑 → (Base‘𝑌) ∈ 𝑈)
3117, 27, 30wunxp 10742 . . . . . 6 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
3217, 21, 31wunop 10740 . . . . 5 (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
33 df-hom 17251 . . . . . . 7 Hom = Slot 14
3433, 17, 20wunstr 17151 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3517, 31, 31wunxp 10742 . . . . . . . 8 (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
3633, 17, 26wunstr 17151 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
3717, 36wunrn 10747 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
3817, 37wununi 10724 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑋) ∈ 𝑈)
3933, 17, 29wunstr 17151 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
4017, 39wunrn 10747 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
4117, 40wununi 10724 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑌) ∈ 𝑈)
4217, 38, 41wunxp 10742 . . . . . . . . 9 (𝜑 → ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
4317, 42wunpw 10725 . . . . . . . 8 (𝜑 → 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
44 ovssunirn 7449 . . . . . . . . . . . . 13 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ⊆ ran (Hom ‘𝑋)
45 ovssunirn 7449 . . . . . . . . . . . . 13 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ⊆ ran (Hom ‘𝑌)
46 xpss12 5688 . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 14 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ∈ V
49 ovex 7446 . . . . . . . . . . . . . 14 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ∈ V
5048, 49xpex 7750 . . . . . . . . . . . . 13 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ V
5150elpw 4603 . . . . . . . . . . . 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 230 . . . . . . . . . . 11 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
5352rgen2w 3056 . . . . . . . . . 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 2725 . . . . . . . . . . 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 8066 . . . . . . . . . 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 229 . . . . . . . . 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 10745 . . . . . . 7 (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))) ∈ 𝑈)
5913, 58eqeltrid 2829 . . . . . 6 (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
6017, 34, 59wunop 10740 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
61 df-cco 17252 . . . . . . 7 comp = Slot 15
6261, 17, 20wunstr 17151 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
6317, 35, 31wunxp 10742 . . . . . . 7 (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
6461, 17, 26wunstr 17151 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
6517, 64wunrn 10747 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
6617, 65wununi 10724 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
6717, 66wunrn 10747 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
6817, 67wununi 10724 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑋) ∈ 𝑈)
6917, 68wunpw 10725 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑋) ∈ 𝑈)
7061, 17, 29wunstr 17151 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
7117, 70wunrn 10747 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
7217, 71wununi 10724 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
7317, 72wunrn 10747 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
7417, 73wununi 10724 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
7517, 74wunpw 10725 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
7617, 69, 75wunxp 10742 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ 𝑈)
7717, 59wunrn 10747 . . . . . . . . . 10 (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
7817, 77wununi 10724 . . . . . . . . 9 (𝜑 ran (Hom ‘𝑇) ∈ 𝑈)
7917, 78, 78wunxp 10742 . . . . . . . 8 (𝜑 → ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ 𝑈)
8017, 76, 79wunpm 10743 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))) ∈ 𝑈)
81 fvex 6903 . . . . . . . . . . . . . . . . 17 (comp‘𝑋) ∈ V
8281rnex 7912 . . . . . . . . . . . . . . . 16 ran (comp‘𝑋) ∈ V
8382uniex 7741 . . . . . . . . . . . . . . 15 ran (comp‘𝑋) ∈ V
8483rnex 7912 . . . . . . . . . . . . . 14 ran ran (comp‘𝑋) ∈ V
8584uniex 7741 . . . . . . . . . . . . 13 ran ran (comp‘𝑋) ∈ V
8685pwex 5375 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑋) ∈ V
87 fvex 6903 . . . . . . . . . . . . . . . . 17 (comp‘𝑌) ∈ V
8887rnex 7912 . . . . . . . . . . . . . . . 16 ran (comp‘𝑌) ∈ V
8988uniex 7741 . . . . . . . . . . . . . . 15 ran (comp‘𝑌) ∈ V
9089rnex 7912 . . . . . . . . . . . . . 14 ran ran (comp‘𝑌) ∈ V
9190uniex 7741 . . . . . . . . . . . . 13 ran ran (comp‘𝑌) ∈ V
9291pwex 5375 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑌) ∈ V
9386, 92xpex 7750 . . . . . . . . . . 11 (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ V
94 fvex 6903 . . . . . . . . . . . . . 14 (Hom ‘𝑇) ∈ V
9594rnex 7912 . . . . . . . . . . . . 13 ran (Hom ‘𝑇) ∈ V
9695uniex 7741 . . . . . . . . . . . 12 ran (Hom ‘𝑇) ∈ V
9796, 96xpex 7750 . . . . . . . . . . 11 ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ V
98 ovssunirn 7449 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))
99 ovssunirn 7449 . . . . . . . . . . . . . . . . 17 (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋)
100 rnss 5936 . . . . . . . . . . . . . . . . 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 4912 . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . 15 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran ran (comp‘𝑋)
104 ovex 7446 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ V
105104elpw 4603 . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . 14 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋)
107 ovssunirn 7449 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))
108 ovssunirn 7449 . . . . . . . . . . . . . . . . 17 (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌)
109 rnss 5936 . . . . . . . . . . . . . . . . 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 4912 . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . 15 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran ran (comp‘𝑌)
113 ovex 7446 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ V
114113elpw 4603 . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . 14 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌)
116 opelxpi 5710 . . . . . . . . . . . . . 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 3056 . . . . . . . . . . . 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 2725 . . . . . . . . . . . . 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 8066 . . . . . . . . . . . 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 229 . . . . . . . . . . 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 7449 . . . . . . . . . . . 12 ((2nd𝑥)(Hom ‘𝑇)𝑦) ⊆ ran (Hom ‘𝑇)
123 fvssunirn 6923 . . . . . . . . . . . 12 ((Hom ‘𝑇)‘𝑥) ⊆ ran (Hom ‘𝑇)
124 xpss12 5688 . . . . . . . . . . . 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 8857 . . . . . . . . . . 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 3056 . . . . . . . . 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 2725 . . . . . . . . . 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 8066 . . . . . . . . 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 229 . . . . . . . 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 10745 . . . . . 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 10740 . . . . 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 10729 . . . 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 2825 . . 3 (𝜑𝑇𝑈)
13725elin2d 4194 . . . 4 (𝜑𝑋 ∈ Cat)
13828elin2d 4194 . . . 4 (𝜑𝑌 ∈ Cat)
1391, 137, 138xpccat 18175 . . 3 (𝜑𝑇 ∈ Cat)
140136, 139elind 4189 . 2 (𝜑𝑇 ∈ (𝑈 ∩ Cat))
141140, 24eleqtrrd 2828 1 (𝜑𝑇𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3051  Vcvv 3463  cin 3940  wss 3941  𝒫 cpw 4599  {ctp 4629  cop 4631   cuni 4904   × cxp 5671  ran crn 5674  wf 6539  cfv 6543  (class class class)co 7413  cmpo 7415  ωcom 7865  1st c1st 7985  2nd c2nd 7986  pm cpm 8839  WUnicwun 10718  1c1 11134  4c4 12294  5c5 12295  cdc 12702  ndxcnx 17156  Basecbs 17174  Hom chom 17238  compcco 17239  Catccat 17638  CatCatccatc 18081   ×c cxpc 18153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-ec 8720  df-qs 8724  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-wun 10720  df-ni 10890  df-pli 10891  df-mi 10892  df-lti 10893  df-plpq 10926  df-mpq 10927  df-ltpq 10928  df-enq 10929  df-nq 10930  df-erq 10931  df-plq 10932  df-mq 10933  df-1nq 10934  df-rq 10935  df-ltnq 10936  df-np 10999  df-plp 11001  df-ltp 11003  df-enr 11073  df-nr 11074  df-c 11139  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-slot 17145  df-ndx 17157  df-base 17175  df-hom 17251  df-cco 17252  df-cat 17642  df-cid 17643  df-catc 18082  df-xpc 18157
This theorem is referenced by: (None)
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