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Theorem funcoppc 17761
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 2736 . . 3 (Base‘𝐶) = (Base‘𝐶)
31, 2oppcbas 17599 . 2 (Base‘𝐶) = (Base‘𝑂)
4 funcoppc.p . . 3 𝑃 = (oppCat‘𝐷)
5 eqid 2736 . . 3 (Base‘𝐷) = (Base‘𝐷)
64, 5oppcbas 17599 . 2 (Base‘𝐷) = (Base‘𝑃)
7 eqid 2736 . 2 (Hom ‘𝑂) = (Hom ‘𝑂)
8 eqid 2736 . 2 (Hom ‘𝑃) = (Hom ‘𝑃)
9 eqid 2736 . 2 (Id‘𝑂) = (Id‘𝑂)
10 eqid 2736 . 2 (Id‘𝑃) = (Id‘𝑃)
11 eqid 2736 . 2 (comp‘𝑂) = (comp‘𝑂)
12 eqid 2736 . 2 (comp‘𝑃) = (comp‘𝑃)
13 funcoppc.f . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
14 df-br 5106 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1513, 14sylib 217 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16 funcrcl 17749 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1715, 16syl 17 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1817simpld 495 . . 3 (𝜑𝐶 ∈ Cat)
191oppccat 17604 . . 3 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2018, 19syl 17 . 2 (𝜑𝑂 ∈ Cat)
214oppccat 17604 . . 3 (𝐷 ∈ Cat → 𝑃 ∈ Cat)
2217, 21simpl2im 504 . 2 (𝜑𝑃 ∈ Cat)
232, 5, 13funcf1 17752 . 2 (𝜑𝐹:(Base‘𝐶)⟶(Base‘𝐷))
242, 13funcfn2 17755 . . 3 (𝜑𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
25 tposfn 8186 . . 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 2736 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
28 eqid 2736 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
2913adantr 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
30 simprr 771 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
31 simprl 769 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
322, 27, 28, 29, 30, 31funcf2 17754 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
33 ovtpos 8172 . . . . 5 (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
3433feq1i 6659 . . . 4 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
3527, 1oppchom 17596 . . . . 5 (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
3628, 4oppchom 17596 . . . . 5 ((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) = ((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥))
3735, 36feq23i 6662 . . . 4 ((𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3834, 37bitri 274 . . 3 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3932, 38sylibr 233 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
40 eqid 2736 . . . 4 (Id‘𝐶) = (Id‘𝐶)
41 eqid 2736 . . . 4 (Id‘𝐷) = (Id‘𝐷)
4213adantr 481 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
43 simpr 485 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
442, 40, 41, 42, 43funcid 17756 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
45 ovtpos 8172 . . . . 5 (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
4645a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
471, 40oppcid 17603 . . . . . . 7 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
4818, 47syl 17 . . . . . 6 (𝜑 → (Id‘𝑂) = (Id‘𝐶))
4948adantr 481 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
5049fveq1d 6844 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
5146, 50fveq12d 6849 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
524, 41oppcid 17603 . . . . . 6 (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
5317, 52simpl2im 504 . . . . 5 (𝜑 → (Id‘𝑃) = (Id‘𝐷))
5453adantr 481 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
5554fveq1d 6844 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
5644, 51, 553eqtr4d 2786 . 2 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹𝑥)))
57 eqid 2736 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
58 eqid 2736 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
59133ad2ant1 1133 . . . . 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 1208 . . . . 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 1207 . . . . 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 1206 . . . . 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 1202 . . . . . 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 17596 . . . . . 6 (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
6563, 64eleqtrdi 2848 . . . . 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 1201 . . . . . 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 2848 . . . . 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 17757 . . . 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 17598 . . . . 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 6846 . . . 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 1133 . . . . . 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, 62ffvelcdmd 7036 . . . . 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, 61ffvelcdmd 7036 . . . . 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, 60ffvelcdmd 7036 . . . . 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 17598 . . . 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 2786 . . 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 8172 . . . 4 (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
7877fveq1i 6843 . . 3 ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
79 ovtpos 8172 . . . . 5 (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
8079fveq1i 6843 . . . 4 ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
8133fveq1i 6843 . . . 4 ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
8280, 81oveq12i 7369 . . 3 (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓))
8376, 78, 823eqtr4g 2801 . 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 17751 1 (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105   × cxp 5631   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  tpos ctpos 8156  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  oppCatcoppc 17591   Func cfunc 17740
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-oppc 17592  df-func 17744
This theorem is referenced by:  fulloppc  17809  fthoppc  17810  yonedalem1  18161  yonedalem21  18162  yonedalem22  18167
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