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Theorem catcxpccl 16768
 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 2621 . . . . 5 (Base‘𝑋) = (Base‘𝑋)
3 eqid 2621 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
4 eqid 2621 . . . . 5 (Hom ‘𝑋) = (Hom ‘𝑋)
5 eqid 2621 . . . . 5 (Hom ‘𝑌) = (Hom ‘𝑌)
6 eqid 2621 . . . . 5 (comp‘𝑋) = (comp‘𝑋)
7 eqid 2621 . . . . 5 (comp‘𝑌) = (comp‘𝑌)
8 catcxpccl.x . . . . 5 (𝜑𝑋𝐵)
9 catcxpccl.y . . . . 5 (𝜑𝑌𝐵)
10 eqidd 2622 . . . . 5 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
111, 2, 3xpcbas 16739 . . . . . . 7 ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12 eqid 2621 . . . . . . 7 (Hom ‘𝑇) = (Hom ‘𝑇)
131, 11, 4, 5, 12xpchomfval 16740 . . . . . 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 2622 . . . . 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 16738 . . . 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 15786 . . . . . . 7 Base = Slot 1
19 catcxpccl.1 . . . . . . . 8 (𝜑 → ω ∈ 𝑈)
2017, 19wunndx 15800 . . . . . . 7 (𝜑 → ndx ∈ 𝑈)
2118, 17, 20wunstr 15803 . . . . . 6 (𝜑 → (Base‘ndx) ∈ 𝑈)
22 inss1 3811 . . . . . . . . 9 (𝑈 ∩ Cat) ⊆ 𝑈
23 catcxpccl.c . . . . . . . . . . 11 𝐶 = (CatCat‘𝑈)
24 catcxpccl.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
2523, 24, 17catcbas 16668 . . . . . . . . . 10 (𝜑𝐵 = (𝑈 ∩ Cat))
268, 25eleqtrd 2700 . . . . . . . . 9 (𝜑𝑋 ∈ (𝑈 ∩ Cat))
2722, 26sseldi 3581 . . . . . . . 8 (𝜑𝑋𝑈)
2818, 17, 27wunstr 15803 . . . . . . 7 (𝜑 → (Base‘𝑋) ∈ 𝑈)
299, 25eleqtrd 2700 . . . . . . . . 9 (𝜑𝑌 ∈ (𝑈 ∩ Cat))
3022, 29sseldi 3581 . . . . . . . 8 (𝜑𝑌𝑈)
3118, 17, 30wunstr 15803 . . . . . . 7 (𝜑 → (Base‘𝑌) ∈ 𝑈)
3217, 28, 31wunxp 9490 . . . . . 6 (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
3317, 21, 32wunop 9488 . . . . 5 (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
34 df-hom 15887 . . . . . . 7 Hom = Slot 14
3534, 17, 20wunstr 15803 . . . . . 6 (𝜑 → (Hom ‘ndx) ∈ 𝑈)
3617, 32, 32wunxp 9490 . . . . . . . 8 (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
3734, 17, 27wunstr 15803 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
3817, 37wunrn 9495 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
3917, 38wununi 9472 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑋) ∈ 𝑈)
4034, 17, 30wunstr 15803 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
4117, 40wunrn 9495 . . . . . . . . . . 11 (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
4217, 41wununi 9472 . . . . . . . . . 10 (𝜑 ran (Hom ‘𝑌) ∈ 𝑈)
4317, 39, 42wunxp 9490 . . . . . . . . 9 (𝜑 → ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
4417, 43wunpw 9473 . . . . . . . 8 (𝜑 → 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌)) ∈ 𝑈)
45 ovssunirn 6634 . . . . . . . . . . . . 13 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ⊆ ran (Hom ‘𝑋)
46 ovssunirn 6634 . . . . . . . . . . . . 13 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ⊆ ran (Hom ‘𝑌)
47 xpss12 5186 . . . . . . . . . . . . 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 707 . . . . . . . . . . . 12 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ⊆ ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
49 ovex 6632 . . . . . . . . . . . . . 14 ((1st𝑢)(Hom ‘𝑋)(1st𝑣)) ∈ V
50 ovex 6632 . . . . . . . . . . . . . 14 ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)) ∈ V
5149, 50xpex 6915 . . . . . . . . . . . . 13 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ V
5251elpw 4136 . . . . . . . . . . . 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 221 . . . . . . . . . . 11 (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣))) ∈ 𝒫 ( ran (Hom ‘𝑋) × ran (Hom ‘𝑌))
5453rgen2w 2920 . . . . . . . . . 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 2621 . . . . . . . . . . 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 7182 . . . . . . . . . 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 220 . . . . . . . . 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 9493 . . . . . . 7 (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st𝑢)(Hom ‘𝑋)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑌)(2nd𝑣)))) ∈ 𝑈)
6013, 59syl5eqel 2702 . . . . . 6 (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
6117, 35, 60wunop 9488 . . . . 5 (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
62 df-cco 15888 . . . . . . 7 comp = Slot 15
6362, 17, 20wunstr 15803 . . . . . 6 (𝜑 → (comp‘ndx) ∈ 𝑈)
6417, 36, 32wunxp 9490 . . . . . . 7 (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
6562, 17, 27wunstr 15803 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑋) ∈ 𝑈)
6617, 65wunrn 9495 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
6717, 66wununi 9472 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑋) ∈ 𝑈)
6817, 67wunrn 9495 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑋) ∈ 𝑈)
6917, 68wununi 9472 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑋) ∈ 𝑈)
7017, 69wunpw 9473 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑋) ∈ 𝑈)
7162, 17, 30wunstr 15803 . . . . . . . . . . . . . 14 (𝜑 → (comp‘𝑌) ∈ 𝑈)
7217, 71wunrn 9495 . . . . . . . . . . . . 13 (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
7317, 72wununi 9472 . . . . . . . . . . . 12 (𝜑 ran (comp‘𝑌) ∈ 𝑈)
7417, 73wunrn 9495 . . . . . . . . . . 11 (𝜑 → ran ran (comp‘𝑌) ∈ 𝑈)
7517, 74wununi 9472 . . . . . . . . . 10 (𝜑 ran ran (comp‘𝑌) ∈ 𝑈)
7617, 75wunpw 9473 . . . . . . . . 9 (𝜑 → 𝒫 ran ran (comp‘𝑌) ∈ 𝑈)
7717, 70, 76wunxp 9490 . . . . . . . 8 (𝜑 → (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ 𝑈)
7817, 60wunrn 9495 . . . . . . . . . 10 (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
7917, 78wununi 9472 . . . . . . . . 9 (𝜑 ran (Hom ‘𝑇) ∈ 𝑈)
8017, 79, 79wunxp 9490 . . . . . . . 8 (𝜑 → ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ 𝑈)
8117, 77, 80wunpm 9491 . . . . . . 7 (𝜑 → ((𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ↑pm ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))) ∈ 𝑈)
82 fvex 6158 . . . . . . . . . . . . . . . . 17 (comp‘𝑋) ∈ V
8382rnex 7047 . . . . . . . . . . . . . . . 16 ran (comp‘𝑋) ∈ V
8483uniex 6906 . . . . . . . . . . . . . . 15 ran (comp‘𝑋) ∈ V
8584rnex 7047 . . . . . . . . . . . . . 14 ran ran (comp‘𝑋) ∈ V
8685uniex 6906 . . . . . . . . . . . . 13 ran ran (comp‘𝑋) ∈ V
8786pwex 4808 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑋) ∈ V
88 fvex 6158 . . . . . . . . . . . . . . . . 17 (comp‘𝑌) ∈ V
8988rnex 7047 . . . . . . . . . . . . . . . 16 ran (comp‘𝑌) ∈ V
9089uniex 6906 . . . . . . . . . . . . . . 15 ran (comp‘𝑌) ∈ V
9190rnex 7047 . . . . . . . . . . . . . 14 ran ran (comp‘𝑌) ∈ V
9291uniex 6906 . . . . . . . . . . . . 13 ran ran (comp‘𝑌) ∈ V
9392pwex 4808 . . . . . . . . . . . 12 𝒫 ran ran (comp‘𝑌) ∈ V
9487, 93xpex 6915 . . . . . . . . . . 11 (𝒫 ran ran (comp‘𝑋) × 𝒫 ran ran (comp‘𝑌)) ∈ V
95 fvex 6158 . . . . . . . . . . . . . 14 (Hom ‘𝑇) ∈ V
9695rnex 7047 . . . . . . . . . . . . 13 ran (Hom ‘𝑇) ∈ V
9796uniex 6906 . . . . . . . . . . . 12 ran (Hom ‘𝑇) ∈ V
9897, 97xpex 6915 . . . . . . . . . . 11 ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇)) ∈ V
99 ovssunirn 6634 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))
100 ovssunirn 6634 . . . . . . . . . . . . . . . . 17 (⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦)) ⊆ ran (comp‘𝑋)
101 rnss 5314 . . . . . . . . . . . . . . . . 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 4424 . . . . . . . . . . . . . . . . 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 3592 . . . . . . . . . . . . . . 15 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ⊆ ran ran (comp‘𝑋)
105 ovex 6632 . . . . . . . . . . . . . . . 16 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ V
106105elpw 4136 . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . 14 ((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑋)(1st𝑦))(1st𝑓)) ∈ 𝒫 ran ran (comp‘𝑋)
108 ovssunirn 6634 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))
109 ovssunirn 6634 . . . . . . . . . . . . . . . . 17 (⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦)) ⊆ ran (comp‘𝑌)
110 rnss 5314 . . . . . . . . . . . . . . . . 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 4424 . . . . . . . . . . . . . . . . 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 3592 . . . . . . . . . . . . . . 15 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ⊆ ran ran (comp‘𝑌)
114 ovex 6632 . . . . . . . . . . . . . . . 16 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ V
115114elpw 4136 . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . 14 ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑌)(2nd𝑦))(2nd𝑓)) ∈ 𝒫 ran ran (comp‘𝑌)
117 opelxpi 5108 . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . 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 2920 . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 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 7182 . . . . . . . . . . . 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 220 . . . . . . . . . . 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 6634 . . . . . . . . . . . 12 ((2nd𝑥)(Hom ‘𝑇)𝑦) ⊆ ran (Hom ‘𝑇)
124 fvssunirn 6174 . . . . . . . . . . . 12 ((Hom ‘𝑇)‘𝑥) ⊆ ran (Hom ‘𝑇)
125 xpss12 5186 . . . . . . . . . . . 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 707 . . . . . . . . . . 11 (((2nd𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ ( ran (Hom ‘𝑇) × ran (Hom ‘𝑇))
127 elpm2r 7819 . . . . . . . . . . 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 708 . . . . . . . . . 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 2920 . . . . . . . . 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 2621 . . . . . . . . . 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 7182 . . . . . . . . 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 220 . . . . . . . 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 9493 . . . . . 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 9488 . . . . 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 9477 . . . 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 2698 . . 3 (𝜑𝑇𝑈)
138 inss2 3812 . . . . 5 (𝑈 ∩ Cat) ⊆ Cat
139138, 26sseldi 3581 . . . 4 (𝜑𝑋 ∈ Cat)
140138, 29sseldi 3581 . . . 4 (𝜑𝑌 ∈ Cat)
1411, 139, 140xpccat 16751 . . 3 (𝜑𝑇 ∈ Cat)
142137, 141elind 3776 . 2 (𝜑𝑇 ∈ (𝑈 ∩ Cat))
143142, 25eleqtrrd 2701 1 (𝜑𝑇𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ∀wral 2907  Vcvv 3186   ∩ cin 3554   ⊆ wss 3555  𝒫 cpw 4130  {ctp 4152  ⟨cop 4154  ∪ cuni 4402   × cxp 5072  ran crn 5075  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  ωcom 7012  1st c1st 7111  2nd c2nd 7112   ↑pm cpm 7803  WUnicwun 9466  1c1 9881  4c4 11016  5c5 11017  ;cdc 11437  ndxcnx 15778  Basecbs 15781  Hom chom 15873  compcco 15874  Catccat 16246  CatCatccatc 16665   ×c cxpc 16729 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wun 9468  df-ni 9638  df-pli 9639  df-mi 9640  df-lti 9641  df-plpq 9674  df-mpq 9675  df-ltpq 9676  df-enq 9677  df-nq 9678  df-erq 9679  df-plq 9680  df-mq 9681  df-1nq 9682  df-rq 9683  df-ltnq 9684  df-np 9747  df-plp 9749  df-ltp 9751  df-enr 9821  df-nr 9822  df-c 9886  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-catc 16666  df-xpc 16733 This theorem is referenced by: (None)
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