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Theorem rescabs 16316
 Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
rescabs.c (𝜑𝐶𝑉)
rescabs.h (𝜑𝐻 Fn (𝑆 × 𝑆))
rescabs.j (𝜑𝐽 Fn (𝑇 × 𝑇))
rescabs.s (𝜑𝑆𝑊)
rescabs.t (𝜑𝑇𝑆)
Assertion
Ref Expression
rescabs (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))

Proof of Theorem rescabs
StepHypRef Expression
1 eqid 2610 . . . 4 (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽)
2 ovex 6577 . . . . 5 ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
32a1i 11 . . . 4 (𝜑 → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
4 rescabs.s . . . . 5 (𝜑𝑆𝑊)
5 rescabs.t . . . . 5 (𝜑𝑇𝑆)
64, 5ssexd 4733 . . . 4 (𝜑𝑇 ∈ V)
7 rescabs.j . . . 4 (𝜑𝐽 Fn (𝑇 × 𝑇))
81, 3, 6, 7rescval2 16311 . . 3 (𝜑 → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9 simpr 476 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶s 𝑆)) ⊆ 𝑇)
102a1i 11 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
116adantr 480 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
12 eqid 2610 . . . . . . . 8 (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇)
13 baseid 15747 . . . . . . . . 9 Base = Slot (Base‘ndx)
14 1re 9918 . . . . . . . . . . 11 1 ∈ ℝ
15 1nn 10908 . . . . . . . . . . . 12 1 ∈ ℕ
16 4nn0 11188 . . . . . . . . . . . 12 4 ∈ ℕ0
17 1nn0 11185 . . . . . . . . . . . 12 1 ∈ ℕ0
18 1lt10 11557 . . . . . . . . . . . 12 1 < 10
1915, 16, 17, 18declti 11422 . . . . . . . . . . 11 1 < 14
2014, 19ltneii 10029 . . . . . . . . . 10 1 ≠ 14
21 basendx 15751 . . . . . . . . . . 11 (Base‘ndx) = 1
22 homndx 15897 . . . . . . . . . . 11 (Hom ‘ndx) = 14
2321, 22neeq12i 2848 . . . . . . . . . 10 ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ 14)
2420, 23mpbir 220 . . . . . . . . 9 (Base‘ndx) ≠ (Hom ‘ndx)
2513, 24setsnid 15743 . . . . . . . 8 (Base‘(𝐶s 𝑆)) = (Base‘((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
2612, 25ressid2 15755 . . . . . . 7 (((Base‘(𝐶s 𝑆)) ⊆ 𝑇 ∧ ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
279, 10, 11, 26syl3anc 1318 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
2827oveq1d 6564 . . . . 5 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩))
29 ovex 6577 . . . . . 6 (𝐶s 𝑆) ∈ V
30 xpexg 6858 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇 ∈ V) → (𝑇 × 𝑇) ∈ V)
316, 6, 30syl2anc 691 . . . . . . . 8 (𝜑 → (𝑇 × 𝑇) ∈ V)
32 fnex 6386 . . . . . . . 8 ((𝐽 Fn (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ∈ V) → 𝐽 ∈ V)
337, 31, 32syl2anc 691 . . . . . . 7 (𝜑𝐽 ∈ V)
3433adantr 480 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
35 setsabs 15730 . . . . . 6 (((𝐶s 𝑆) ∈ V ∧ 𝐽 ∈ V) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
3629, 34, 35sylancr 694 . . . . 5 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
37 eqid 2610 . . . . . . . . . . . . . 14 (𝐶s 𝑆) = (𝐶s 𝑆)
38 eqid 2610 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
3937, 38ressbas 15757 . . . . . . . . . . . . 13 (𝑆𝑊 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑆)))
404, 39syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑆)))
4140sseq1d 3595 . . . . . . . . . . 11 (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶s 𝑆)) ⊆ 𝑇))
4241biimpar 501 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇)
43 inss2 3796 . . . . . . . . . . 11 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
4443a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
4542, 44ssind 3799 . . . . . . . . 9 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶)))
465adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇𝑆)
47 ssrin 3800 . . . . . . . . . 10 (𝑇𝑆 → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
4846, 47syl 17 . . . . . . . . 9 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
4945, 48eqssd 3585 . . . . . . . 8 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
5049oveq2d 6565 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
514adantr 480 . . . . . . . 8 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑆𝑊)
5238ressinbas 15763 . . . . . . . 8 (𝑆𝑊 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
5351, 52syl 17 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
5438ressinbas 15763 . . . . . . . 8 (𝑇 ∈ V → (𝐶s 𝑇) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
5511, 54syl 17 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑇) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
5650, 53, 553eqtr4d 2654 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) = (𝐶s 𝑇))
5756oveq1d 6564 . . . . 5 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
5828, 36, 573eqtrd 2648 . . . 4 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
59 simpr 476 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇)
602a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
616adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
6212, 25ressval2 15756 . . . . . . . 8 ((¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇 ∧ ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V ∧ 𝑇 ∈ V) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
6359, 60, 61, 62syl3anc 1318 . . . . . . 7 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
6429a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) ∈ V)
6524necomi 2836 . . . . . . . . 9 (Hom ‘ndx) ≠ (Base‘ndx)
6665a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (Hom ‘ndx) ≠ (Base‘ndx))
67 rescabs.h . . . . . . . . . 10 (𝜑𝐻 Fn (𝑆 × 𝑆))
68 xpexg 6858 . . . . . . . . . . 11 ((𝑆𝑊𝑆𝑊) → (𝑆 × 𝑆) ∈ V)
694, 4, 68syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝑆 × 𝑆) ∈ V)
70 fnex 6386 . . . . . . . . . 10 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
7167, 69, 70syl2anc 691 . . . . . . . . 9 (𝜑𝐻 ∈ V)
7271adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
73 fvex 6113 . . . . . . . . . 10 (Base‘(𝐶s 𝑆)) ∈ V
7473inex2 4728 . . . . . . . . 9 (𝑇 ∩ (Base‘(𝐶s 𝑆))) ∈ V
7574a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘(𝐶s 𝑆))) ∈ V)
76 fvex 6113 . . . . . . . . 9 (Hom ‘ndx) ∈ V
77 fvex 6113 . . . . . . . . 9 (Base‘ndx) ∈ V
7876, 77setscom 15731 . . . . . . . 8 ((((𝐶s 𝑆) ∈ V ∧ (Hom ‘ndx) ≠ (Base‘ndx)) ∧ (𝐻 ∈ V ∧ (𝑇 ∩ (Base‘(𝐶s 𝑆))) ∈ V)) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) = (((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩))
7964, 66, 72, 75, 78syl22anc 1319 . . . . . . 7 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) = (((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩))
80 eqid 2610 . . . . . . . . . . 11 ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) ↾s 𝑇)
81 eqid 2610 . . . . . . . . . . 11 (Base‘(𝐶s 𝑆)) = (Base‘(𝐶s 𝑆))
8280, 81ressval2 15756 . . . . . . . . . 10 ((¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇 ∧ (𝐶s 𝑆) ∈ V ∧ 𝑇 ∈ V) → ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
8359, 64, 61, 82syl3anc 1318 . . . . . . . . 9 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
844adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑆𝑊)
855adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇𝑆)
86 ressabs 15766 . . . . . . . . . 10 ((𝑆𝑊𝑇𝑆) → ((𝐶s 𝑆) ↾s 𝑇) = (𝐶s 𝑇))
8784, 85, 86syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) ↾s 𝑇) = (𝐶s 𝑇))
8883, 87eqtr3d 2646 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) = (𝐶s 𝑇))
8988oveq1d 6564 . . . . . . 7 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9063, 79, 893eqtrd 2648 . . . . . 6 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9190oveq1d 6564 . . . . 5 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = (((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩))
92 ovex 6577 . . . . . 6 (𝐶s 𝑇) ∈ V
9333adantr 480 . . . . . 6 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
94 setsabs 15730 . . . . . 6 (((𝐶s 𝑇) ∈ V ∧ 𝐽 ∈ V) → (((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9592, 93, 94sylancr 694 . . . . 5 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9691, 95eqtrd 2644 . . . 4 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9758, 96pm2.61dan 828 . . 3 (𝜑 → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
988, 97eqtrd 2644 . 2 (𝜑 → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
99 eqid 2610 . . . 4 (𝐶cat 𝐻) = (𝐶cat 𝐻)
100 rescabs.c . . . 4 (𝜑𝐶𝑉)
10199, 100, 4, 67rescval2 16311 . . 3 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
102101oveq1d 6564 . 2 (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
103 eqid 2610 . . 3 (𝐶cat 𝐽) = (𝐶cat 𝐽)
104103, 100, 6, 7rescval2 16311 . 2 (𝜑 → (𝐶cat 𝐽) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
10598, 102, 1043eqtr4d 2654 1 (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  1c1 9816  4c4 10949  ;cdc 11369  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾s cress 15696  Hom chom 15779   ↾cat cresc 16291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-resc 16294 This theorem is referenced by:  subsubc  16336  fldc  41875  fldcALTV  41894
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