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Theorem rescabs 16474
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 2620 . . . 4 (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽)
2 ovexd 6665 . . . 4 (𝜑 → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
3 rescabs.s . . . . 5 (𝜑𝑆𝑊)
4 rescabs.t . . . . 5 (𝜑𝑇𝑆)
53, 4ssexd 4796 . . . 4 (𝜑𝑇 ∈ V)
6 rescabs.j . . . 4 (𝜑𝐽 Fn (𝑇 × 𝑇))
71, 2, 5, 6rescval2 16469 . . 3 (𝜑 → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
8 simpr 477 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (Base‘(𝐶s 𝑆)) ⊆ 𝑇)
9 ovexd 6665 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
105adantr 481 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
11 eqid 2620 . . . . . . . 8 (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇)
12 baseid 15900 . . . . . . . . 9 Base = Slot (Base‘ndx)
13 1re 10024 . . . . . . . . . . 11 1 ∈ ℝ
14 1nn 11016 . . . . . . . . . . . 12 1 ∈ ℕ
15 4nn0 11296 . . . . . . . . . . . 12 4 ∈ ℕ0
16 1nn0 11293 . . . . . . . . . . . 12 1 ∈ ℕ0
17 1lt10 11666 . . . . . . . . . . . 12 1 < 10
1814, 15, 16, 17declti 11531 . . . . . . . . . . 11 1 < 14
1913, 18ltneii 10135 . . . . . . . . . 10 1 ≠ 14
20 basendx 15904 . . . . . . . . . . 11 (Base‘ndx) = 1
21 homndx 16055 . . . . . . . . . . 11 (Hom ‘ndx) = 14
2220, 21neeq12i 2857 . . . . . . . . . 10 ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ 14)
2319, 22mpbir 221 . . . . . . . . 9 (Base‘ndx) ≠ (Hom ‘ndx)
2412, 23setsnid 15896 . . . . . . . 8 (Base‘(𝐶s 𝑆)) = (Base‘((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
2511, 24ressid2 15909 . . . . . . 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⟩))
268, 9, 10, 25syl3anc 1324 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
2726oveq1d 6650 . . . . 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⟩))
28 ovex 6663 . . . . . 6 (𝐶s 𝑆) ∈ V
29 xpexg 6945 . . . . . . . . 9 ((𝑇 ∈ V ∧ 𝑇 ∈ V) → (𝑇 × 𝑇) ∈ V)
305, 5, 29syl2anc 692 . . . . . . . 8 (𝜑 → (𝑇 × 𝑇) ∈ V)
31 fnex 6466 . . . . . . . 8 ((𝐽 Fn (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ∈ V) → 𝐽 ∈ V)
326, 30, 31syl2anc 692 . . . . . . 7 (𝜑𝐽 ∈ V)
3332adantr 481 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
34 setsabs 15883 . . . . . 6 (((𝐶s 𝑆) ∈ V ∧ 𝐽 ∈ V) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
3528, 33, 34sylancr 694 . . . . 5 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩))
36 eqid 2620 . . . . . . . . . . . . . 14 (𝐶s 𝑆) = (𝐶s 𝑆)
37 eqid 2620 . . . . . . . . . . . . . 14 (Base‘𝐶) = (Base‘𝐶)
3836, 37ressbas 15911 . . . . . . . . . . . . 13 (𝑆𝑊 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑆)))
393, 38syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑆 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑆)))
4039sseq1d 3624 . . . . . . . . . . 11 (𝜑 → ((𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇 ↔ (Base‘(𝐶s 𝑆)) ⊆ 𝑇))
4140biimpar 502 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ 𝑇)
42 inss2 3826 . . . . . . . . . . 11 (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶)
4342a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (Base‘𝐶))
4441, 43ssind 3829 . . . . . . . . 9 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) ⊆ (𝑇 ∩ (Base‘𝐶)))
454adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇𝑆)
46 ssrin 3830 . . . . . . . . . 10 (𝑇𝑆 → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
4745, 46syl 17 . . . . . . . . 9 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘𝐶)) ⊆ (𝑆 ∩ (Base‘𝐶)))
4844, 47eqssd 3612 . . . . . . . 8 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑆 ∩ (Base‘𝐶)) = (𝑇 ∩ (Base‘𝐶)))
4948oveq2d 6651 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s (𝑆 ∩ (Base‘𝐶))) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
503adantr 481 . . . . . . . 8 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑆𝑊)
5137ressinbas 15917 . . . . . . . 8 (𝑆𝑊 → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
5250, 51syl 17 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) = (𝐶s (𝑆 ∩ (Base‘𝐶))))
5337ressinbas 15917 . . . . . . . 8 (𝑇 ∈ V → (𝐶s 𝑇) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
5410, 53syl 17 . . . . . . 7 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑇) = (𝐶s (𝑇 ∩ (Base‘𝐶))))
5549, 52, 543eqtr4d 2664 . . . . . 6 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) = (𝐶s 𝑇))
5655oveq1d 6650 . . . . 5 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
5727, 35, 563eqtrd 2658 . . . 4 ((𝜑 ∧ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
58 simpr 477 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇)
59 ovexd 6665 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V)
605adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇 ∈ V)
6111, 24ressval2 15910 . . . . . . . 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)))⟩))
6258, 59, 60, 61syl3anc 1324 . . . . . . 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)))⟩))
63 ovexd 6665 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝐶s 𝑆) ∈ V)
6423necomi 2845 . . . . . . . . 9 (Hom ‘ndx) ≠ (Base‘ndx)
6564a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (Hom ‘ndx) ≠ (Base‘ndx))
66 rescabs.h . . . . . . . . . 10 (𝜑𝐻 Fn (𝑆 × 𝑆))
67 xpexg 6945 . . . . . . . . . . 11 ((𝑆𝑊𝑆𝑊) → (𝑆 × 𝑆) ∈ V)
683, 3, 67syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝑆 × 𝑆) ∈ V)
69 fnex 6466 . . . . . . . . . 10 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
7066, 68, 69syl2anc 692 . . . . . . . . 9 (𝜑𝐻 ∈ V)
7170adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐻 ∈ V)
72 fvex 6188 . . . . . . . . . 10 (Base‘(𝐶s 𝑆)) ∈ V
7372inex2 4791 . . . . . . . . 9 (𝑇 ∩ (Base‘(𝐶s 𝑆))) ∈ V
7473a1i 11 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (𝑇 ∩ (Base‘(𝐶s 𝑆))) ∈ V)
75 fvex 6188 . . . . . . . . 9 (Hom ‘ndx) ∈ V
76 fvex 6188 . . . . . . . . 9 (Base‘ndx) ∈ V
7775, 76setscom 15884 . . . . . . . 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⟩))
7863, 65, 71, 74, 77syl22anc 1325 . . . . . . 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⟩))
79 eqid 2620 . . . . . . . . . . 11 ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) ↾s 𝑇)
80 eqid 2620 . . . . . . . . . . 11 (Base‘(𝐶s 𝑆)) = (Base‘(𝐶s 𝑆))
8179, 80ressval2 15910 . . . . . . . . . 10 ((¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇 ∧ (𝐶s 𝑆) ∈ V ∧ 𝑇 ∈ V) → ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
8258, 63, 60, 81syl3anc 1324 . . . . . . . . 9 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) ↾s 𝑇) = ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩))
833adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑆𝑊)
844adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝑇𝑆)
85 ressabs 15920 . . . . . . . . . 10 ((𝑆𝑊𝑇𝑆) → ((𝐶s 𝑆) ↾s 𝑇) = (𝐶s 𝑇))
8683, 84, 85syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) ↾s 𝑇) = (𝐶s 𝑇))
8782, 86eqtr3d 2656 . . . . . . . 8 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((𝐶s 𝑆) sSet ⟨(Base‘ndx), (𝑇 ∩ (Base‘(𝐶s 𝑆)))⟩) = (𝐶s 𝑇))
8887oveq1d 6650 . . . . . . 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⟩))
8962, 78, 883eqtrd 2658 . . . . . 6 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9089oveq1d 6650 . . . . 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⟩))
91 ovex 6663 . . . . . 6 (𝐶s 𝑇) ∈ V
9232adantr 481 . . . . . 6 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → 𝐽 ∈ V)
93 setsabs 15883 . . . . . 6 (((𝐶s 𝑇) ∈ V ∧ 𝐽 ∈ V) → (((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9491, 92, 93sylancr 694 . . . . 5 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → (((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐻⟩) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9590, 94eqtrd 2654 . . . 4 ((𝜑 ∧ ¬ (Base‘(𝐶s 𝑆)) ⊆ 𝑇) → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
9657, 95pm2.61dan 831 . . 3 (𝜑 → ((((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
977, 96eqtrd 2654 . 2 (𝜑 → (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
98 eqid 2620 . . . 4 (𝐶cat 𝐻) = (𝐶cat 𝐻)
99 rescabs.c . . . 4 (𝜑𝐶𝑉)
10098, 99, 3, 66rescval2 16469 . . 3 (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
101100oveq1d 6650 . 2 (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩) ↾cat 𝐽))
102 eqid 2620 . . 3 (𝐶cat 𝐽) = (𝐶cat 𝐽)
103102, 99, 5, 6rescval2 16469 . 2 (𝜑 → (𝐶cat 𝐽) = ((𝐶s 𝑇) sSet ⟨(Hom ‘ndx), 𝐽⟩))
10497, 101, 1033eqtr4d 2664 1 (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  cin 3566  wss 3567  cop 4174   × cxp 5102   Fn wfn 5871  cfv 5876  (class class class)co 6635  1c1 9922  4c4 11057  cdc 11478  ndxcnx 15835   sSet csts 15836  Basecbs 15838  s cress 15839  Hom chom 15933  cat cresc 16449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-hom 15947  df-resc 16452
This theorem is referenced by:  subsubc  16494  fldc  41848  fldcALTV  41866
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