Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcoppc Structured version   Visualization version   GIF version

Theorem funcoppc 16582
 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 2651 . . 3 (Base‘𝐶) = (Base‘𝐶)
31, 2oppcbas 16425 . 2 (Base‘𝐶) = (Base‘𝑂)
4 funcoppc.p . . 3 𝑃 = (oppCat‘𝐷)
5 eqid 2651 . . 3 (Base‘𝐷) = (Base‘𝐷)
64, 5oppcbas 16425 . 2 (Base‘𝐷) = (Base‘𝑃)
7 eqid 2651 . 2 (Hom ‘𝑂) = (Hom ‘𝑂)
8 eqid 2651 . 2 (Hom ‘𝑃) = (Hom ‘𝑃)
9 eqid 2651 . 2 (Id‘𝑂) = (Id‘𝑂)
10 eqid 2651 . 2 (Id‘𝑃) = (Id‘𝑃)
11 eqid 2651 . 2 (comp‘𝑂) = (comp‘𝑂)
12 eqid 2651 . 2 (comp‘𝑃) = (comp‘𝑃)
13 funcoppc.f . . . . . 6 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
14 df-br 4686 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
1513, 14sylib 208 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
16 funcrcl 16570 . . . . 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 16429 . . 3 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
2018, 19syl 17 . 2 (𝜑𝑂 ∈ Cat)
2117simprd 478 . . 3 (𝜑𝐷 ∈ Cat)
224oppccat 16429 . . 3 (𝐷 ∈ Cat → 𝑃 ∈ Cat)
2321, 22syl 17 . 2 (𝜑𝑃 ∈ Cat)
242, 5, 13funcf1 16573 . 2 (𝜑𝐹:(Base‘𝐶)⟶(Base‘𝐷))
252, 13funcfn2 16576 . . 3 (𝜑𝐺 Fn ((Base‘𝐶) × (Base‘𝐶)))
26 tposfn 7426 . . 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 2651 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
29 eqid 2651 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
3013adantr 480 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹(𝐶 Func 𝐷)𝐺)
31 simprr 811 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
32 simprl 809 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
332, 28, 29, 30, 31, 32funcf2 16575 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
34 ovtpos 7412 . . . . 5 (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
3534feq1i 6074 . . . 4 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
3628, 1oppchom 16422 . . . . 5 (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
3729, 4oppchom 16422 . . . . 5 ((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) = ((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥))
3836, 37feq23i 6077 . . . 4 ((𝑦𝐺𝑥):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
3935, 38bitri 264 . . 3 ((𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)) ↔ (𝑦𝐺𝑥):(𝑦(Hom ‘𝐶)𝑥)⟶((𝐹𝑦)(Hom ‘𝐷)(𝐹𝑥)))
4033, 39sylibr 224 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥tpos 𝐺𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶((𝐹𝑥)(Hom ‘𝑃)(𝐹𝑦)))
41 eqid 2651 . . . 4 (Id‘𝐶) = (Id‘𝐶)
42 eqid 2651 . . . 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 16577 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
46 ovtpos 7412 . . . . 5 (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥)
4746a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑥tpos 𝐺𝑥) = (𝑥𝐺𝑥))
481, 41oppcid 16428 . . . . . . 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 6231 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑂)‘𝑥) = ((Id‘𝐶)‘𝑥))
5247, 51fveq12d 6235 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)))
534, 42oppcid 16428 . . . . . 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 6231 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑃)‘(𝐹𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)))
5745, 52, 563eqtr4d 2695 . 2 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥tpos 𝐺𝑥)‘((Id‘𝑂)‘𝑥)) = ((Id‘𝑃)‘(𝐹𝑥)))
58 eqid 2651 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
59 eqid 2651 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
60133ad2ant1 1102 . . . . 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 1116 . . . . 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 1115 . . . . 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 1114 . . . . 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 1110 . . . . . 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 16422 . . . . . 6 (𝑦(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑦)
6664, 65syl6eleq 2740 . . . . 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 1109 . . . . . 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 2740 . . . . 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 16578 . . . 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 16424 . . . . 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 6233 . . . 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 1102 . . . . . 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 6400 . . . . 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 6400 . . . . 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 6400 . . . . 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 16424 . . . 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 2695 . . 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 7412 . . . 4 (𝑥tpos 𝐺𝑧) = (𝑧𝐺𝑥)
7978fveq1i 6230 . . 3 ((𝑥tpos 𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓)) = ((𝑧𝐺𝑥)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑂)𝑧)𝑓))
80 ovtpos 7412 . . . . 5 (𝑦tpos 𝐺𝑧) = (𝑧𝐺𝑦)
8180fveq1i 6230 . . . 4 ((𝑦tpos 𝐺𝑧)‘𝑔) = ((𝑧𝐺𝑦)‘𝑔)
8234fveq1i 6230 . . . 4 ((𝑥tpos 𝐺𝑦)‘𝑓) = ((𝑦𝐺𝑥)‘𝑓)
8381, 82oveq12i 6702 . . 3 (((𝑦tpos 𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑥tpos 𝐺𝑦)‘𝑓)) = (((𝑧𝐺𝑦)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝑃)(𝐹𝑧))((𝑦𝐺𝑥)‘𝑓))
8477, 79, 833eqtr4g 2710 . 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 16572 1 (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   class class class wbr 4685   × cxp 5141   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  tpos ctpos 7396  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  oppCatcoppc 16418   Func cfunc 16561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-oppc 16419  df-func 16565 This theorem is referenced by:  fulloppc  16629  fthoppc  16630  yonedalem1  16959  yonedalem21  16960  yonedalem22  16965
 Copyright terms: Public domain W3C validator