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Theorem funcoppc 17124
 Description: A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
funcoppc.o 𝑂 = (oppCat‘𝐶)
funcoppc.p 𝑃 = (oppCat‘𝐷)
funcoppc.f (𝜑𝐹(𝐶 Func 𝐷)𝐺)
Assertion
Ref Expression
funcoppc (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)

Proof of Theorem funcoppc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcoppc.o . . 3 𝑂 = (oppCat‘𝐶)
2 eqid 2821 . . 3 (Base‘𝐶) = (Base‘𝐶)
31, 2oppcbas 16967 . 2 (Base‘𝐶) = (Base‘𝑂)
4 funcoppc.p . . 3 𝑃 = (oppCat‘𝐷)
5 eqid 2821 . . 3 (Base‘𝐷) = (Base‘𝐷)
64, 5oppcbas 16967 . 2 (Base‘𝐷) = (Base‘𝑃)
7 eqid 2821 . 2 (Hom ‘𝑂) = (Hom ‘𝑂)
8 eqid 2821 . 2 (Hom ‘𝑃) = (Hom ‘𝑃)
9 eqid 2821 . 2 (Id‘𝑂) = (Id‘𝑂)
10 eqid 2821 . 2 (Id‘𝑃) = (Id‘𝑃)
11 eqid 2821 . 2 (comp‘𝑂) = (comp‘𝑂)
12 eqid 2821 . 2 (comp‘𝑃) = (comp‘𝑃)
13 funcoppc.f . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
14 df-br 5040 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1513, 14sylib 221 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16 funcrcl 17112 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1715, 16syl 17 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1817simpld 498 . . 3 (𝜑𝐶 ∈ Cat)
191oppccat 16971 . . 3 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2018, 19syl 17 . 2 (𝜑𝑂 ∈ Cat)
214oppccat 16971 . . 3 (𝐷 ∈ Cat → 𝑃 ∈ Cat)
2217, 21simpl2im 507 . 2 (𝜑𝑃 ∈ Cat)
232, 5, 13funcf1 17115 . 2 (𝜑𝐹:(Base‘𝐶)⟶(Base‘𝐷))
242, 13funcfn2 17118 . . 3 (𝜑𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
25 tposfn 7896 . . 3 (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
2624, 25syl 17 . 2 (𝜑 → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
27 eqid 2821 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
28 eqid 2821 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
2913adantr 484 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
30 simprr 772 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
31 simprl 770 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
322, 27, 28, 29, 30, 31funcf2 17117 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
33 ovtpos 7882 . . . . 5 (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
3433feq1i 6478 . . . 4 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
3527, 1oppchom 16964 . . . . 5 (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
3628, 4oppchom 16964 . . . . 5 ((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) = ((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥))
3735, 36feq23i 6481 . . . 4 ((𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3834, 37bitri 278 . . 3 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3932, 38sylibr 237 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
40 eqid 2821 . . . 4 (Id‘𝐶) = (Id‘𝐶)
41 eqid 2821 . . . 4 (Id‘𝐷) = (Id‘𝐷)
4213adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
43 simpr 488 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
442, 40, 41, 42, 43funcid 17119 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
45 ovtpos 7882 . . . . 5 (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
4645a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
471, 40oppcid 16970 . . . . . . 7 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
4818, 47syl 17 . . . . . 6 (𝜑 → (Id‘𝑂) = (Id‘𝐶))
4948adantr 484 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
5049fveq1d 6645 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
5146, 50fveq12d 6650 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
524, 41oppcid 16970 . . . . . 6 (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
5317, 52simpl2im 507 . . . . 5 (𝜑 → (Id‘𝑃) = (Id‘𝐷))
5453adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
5554fveq1d 6645 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
5644, 51, 553eqtr4d 2866 . 2 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹𝑥)))
57 eqid 2821 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
58 eqid 2821 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
59133ad2ant1 1130 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝐹(𝐶 Func 𝐷)𝐺)
60 simp23 1205 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑧 ∈ (Base‘𝐶))
61 simp22 1204 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑦 ∈ (Base‘𝐶))
62 simp21 1203 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑥 ∈ (Base‘𝐶))
63 simp3r 1199 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))
6427, 1oppchom 16964 . . . . . 6 (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
6563, 64eleqtrdi 2922 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑦))
66 simp3l 1198 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦))
6766, 35eleqtrdi 2922 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
682, 27, 57, 58, 59, 60, 61, 62, 65, 67funcco 17120 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)) = (((𝑦𝐺𝑥)‘𝑓)(⟨(𝐹𝑧), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑥))((𝑧𝐺𝑦)‘𝑔)))
692, 57, 1, 62, 61, 60oppcco 16966 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔))
7069fveq2d 6647 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔)))
71233ad2ant1 1130 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
7271, 62ffvelrnd 6825 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑥) ∈ (Base‘𝐷))
7371, 61ffvelrnd 6825 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑦) ∈ (Base‘𝐷))
7471, 60ffvelrnd 6825 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑧) ∈ (Base‘𝐷))
755, 58, 4, 72, 73, 74oppcco 16966 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓)) = (((𝑦𝐺𝑥)‘𝑓)(⟨(𝐹𝑧), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑥))((𝑧𝐺𝑦)‘𝑔)))
7668, 70, 753eqtr4d 2866 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓)))
77 ovtpos 7882 . . . 4 (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
7877fveq1i 6644 . . 3 ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
79 ovtpos 7882 . . . . 5 (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
8079fveq1i 6644 . . . 4 ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
8133fveq1i 6644 . . . 4 ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
8280, 81oveq12i 7142 . . 3 (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓))
8376, 78, 823eqtr4g 2881 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)))
843, 6, 7, 8, 9, 10, 11, 12, 20, 22, 23, 26, 39, 56, 83isfuncd 17114 1 (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ⟨cop 4546   class class class wbr 5039   × cxp 5526   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130  tpos ctpos 7866  Basecbs 16462  Hom chom 16555  compcco 16556  Catccat 16914  Idccid 16915  oppCatcoppc 16960   Func cfunc 17103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-tpos 7867  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-map 8383  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-hom 16568  df-cco 16569  df-cat 16918  df-cid 16919  df-oppc 16961  df-func 17107 This theorem is referenced by:  fulloppc  17171  fthoppc  17172  yonedalem1  17501  yonedalem21  17502  yonedalem22  17507
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