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Theorem funcoppc 16358
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 2610 . . 3 (Base‘𝐶) = (Base‘𝐶)
31, 2oppcbas 16201 . 2 (Base‘𝐶) = (Base‘𝑂)
4 funcoppc.p . . 3 𝑃 = (oppCat‘𝐷)
5 eqid 2610 . . 3 (Base‘𝐷) = (Base‘𝐷)
64, 5oppcbas 16201 . 2 (Base‘𝐷) = (Base‘𝑃)
7 eqid 2610 . 2 (Hom ‘𝑂) = (Hom ‘𝑂)
8 eqid 2610 . 2 (Hom ‘𝑃) = (Hom ‘𝑃)
9 eqid 2610 . 2 (Id‘𝑂) = (Id‘𝑂)
10 eqid 2610 . 2 (Id‘𝑃) = (Id‘𝑃)
11 eqid 2610 . 2 (comp‘𝑂) = (comp‘𝑂)
12 eqid 2610 . 2 (comp‘𝑃) = (comp‘𝑃)
13 funcoppc.f . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
14 df-br 4584 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1513, 14sylib 207 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16 funcrcl 16346 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1715, 16syl 17 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1817simpld 474 . . 3 (𝜑𝐶 ∈ Cat)
191oppccat 16205 . . 3 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2018, 19syl 17 . 2 (𝜑𝑂 ∈ Cat)
2117simprd 478 . . 3 (𝜑𝐷 ∈ Cat)
224oppccat 16205 . . 3 (𝐷 ∈ Cat → 𝑃 ∈ Cat)
2321, 22syl 17 . 2 (𝜑𝑃 ∈ Cat)
242, 5, 13funcf1 16349 . 2 (𝜑𝐹:(Base‘𝐶)⟶(Base‘𝐷))
252, 13funcfn2 16352 . . 3 (𝜑𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
26 tposfn 7268 . . 3 (𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)) → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
2725, 26syl 17 . 2 (𝜑 → tpos 𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
28 eqid 2610 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
29 eqid 2610 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
3013adantr 480 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
31 simprr 792 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
32 simprl 790 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
332, 28, 29, 30, 31, 32funcf2 16351 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
34 ovtpos 7254 . . . . 5 (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
3534feq1i 5949 . . . 4 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
3628, 1oppchom 16198 . . . . 5 (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
3729, 4oppchom 16198 . . . . 5 ((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) = ((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥))
3836, 37feq23i 5952 . . . 4 ((𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3935, 38bitri 263 . . 3 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
4033, 39sylibr 223 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
41 eqid 2610 . . . 4 (Id‘𝐶) = (Id‘𝐶)
42 eqid 2610 . . . 4 (Id‘𝐷) = (Id‘𝐷)
4313adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹(𝐶 Func 𝐷)𝐺)
44 simpr 476 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
452, 41, 42, 43, 44funcid 16353 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
46 ovtpos 7254 . . . . 5 (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
4746a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
481, 41oppcid 16204 . . . . . . 7 (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
4918, 48syl 17 . . . . . 6 (𝜑 → (Id‘𝑂) = (Id‘𝐶))
5049adantr 480 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑂) = (Id‘𝐶))
5150fveq1d 6105 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
5247, 51fveq12d 6109 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
534, 42oppcid 16204 . . . . . 6 (𝐷 ∈ Cat → (Id‘𝑃) = (Id‘𝐷))
5421, 53syl 17 . . . . 5 (𝜑 → (Id‘𝑃) = (Id‘𝐷))
5554adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (Id‘𝑃) = (Id‘𝐷))
5655fveq1d 6105 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
5745, 52, 563eqtr4d 2654 . 2 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹𝑥)))
58 eqid 2610 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
59 eqid 2610 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
60133ad2ant1 1075 . . . . 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)
61 simp23 1089 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑧 ∈ (Base‘𝐶))
62 simp22 1088 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑦 ∈ (Base‘𝐶))
63 simp21 1087 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → 𝑥 ∈ (Base‘𝐶))
64 simp3r 1083 . . . . . 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))
6528, 1oppchom 16198 . . . . . 6 (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
6664, 65syl6eleq 2698 . . . . 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))
67 simp3l 1082 . . . . . 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))
6867, 36syl6eleq 2698 . . . . 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))
692, 28, 58, 59, 60, 61, 62, 63, 66, 68funcco 16354 . . . 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)))
702, 58, 1, 63, 62, 61oppcco 16200 . . . . 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))
7170fveq2d 6107 . . . 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)))
72243ad2ant1 1075 . . . . . 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))
7372, 63ffvelrnd 6268 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑥) ∈ (Base‘𝐷))
7472, 62ffvelrnd 6268 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑦) ∈ (Base‘𝐷))
7572, 61ffvelrnd 6268 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑂)𝑧))) → (𝐹𝑧) ∈ (Base‘𝐷))
765, 59, 4, 73, 74, 75oppcco 16200 . . . 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)))
7769, 71, 763eqtr4d 2654 . . 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)))
78 ovtpos 7254 . . . 4 (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
7978fveq1i 6104 . . 3 ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
80 ovtpos 7254 . . . . 5 (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
8180fveq1i 6104 . . . 4 ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
8234fveq1i 6104 . . . 4 ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
8381, 82oveq12i 6561 . . 3 (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓))
8477, 79, 833eqtr4g 2669 . 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)))
853, 6, 7, 8, 9, 10, 11, 12, 20, 23, 24, 27, 40, 57, 84isfuncd 16348 1 (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583   × cxp 5036   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  tpos ctpos 7238  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  oppCatcoppc 16194   Func cfunc 16337
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-rmo 2904  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-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-ixp 7795  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-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-oppc 16195  df-func 16341
This theorem is referenced by:  fulloppc  16405  fthoppc  16406  yonedalem1  16735  yonedalem21  16736  yonedalem22  16741
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