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Theorem funcco 16578
Description: A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcco.b 𝐵 = (Base‘𝐷)
funcco.h 𝐻 = (Hom ‘𝐷)
funcco.o · = (comp‘𝐷)
funcco.O 𝑂 = (comp‘𝐸)
funcco.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funcco.x (𝜑𝑋𝐵)
funcco.y (𝜑𝑌𝐵)
funcco.z (𝜑𝑍𝐵)
funcco.m (𝜑𝑀 ∈ (𝑋𝐻𝑌))
funcco.n (𝜑𝑁 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
funcco (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀)))

Proof of Theorem funcco
Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcco.f . . . 4 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
2 funcco.b . . . . 5 𝐵 = (Base‘𝐷)
3 eqid 2651 . . . . 5 (Base‘𝐸) = (Base‘𝐸)
4 funcco.h . . . . 5 𝐻 = (Hom ‘𝐷)
5 eqid 2651 . . . . 5 (Hom ‘𝐸) = (Hom ‘𝐸)
6 eqid 2651 . . . . 5 (Id‘𝐷) = (Id‘𝐷)
7 eqid 2651 . . . . 5 (Id‘𝐸) = (Id‘𝐸)
8 funcco.o . . . . 5 · = (comp‘𝐷)
9 funcco.O . . . . 5 𝑂 = (comp‘𝐸)
10 df-br 4686 . . . . . . . 8 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
111, 10sylib 208 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12 funcrcl 16570 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . 6 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simpld 474 . . . . 5 (𝜑𝐷 ∈ Cat)
1513simprd 478 . . . . 5 (𝜑𝐸 ∈ Cat)
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 16571 . . . 4 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))(Hom ‘𝐸)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
171, 16mpbid 222 . . 3 (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))(Hom ‘𝐸)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
1817simp3d 1095 . 2 (𝜑 → ∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
19 funcco.x . . 3 (𝜑𝑋𝐵)
20 funcco.y . . . . . 6 (𝜑𝑌𝐵)
2120adantr 480 . . . . 5 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
22 funcco.z . . . . . . 7 (𝜑𝑍𝐵)
2322ad2antrr 762 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍𝐵)
24 funcco.m . . . . . . . . 9 (𝜑𝑀 ∈ (𝑋𝐻𝑌))
2524ad3antrrr 766 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑀 ∈ (𝑋𝐻𝑌))
26 simpllr 815 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
27 simplr 807 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
2826, 27oveq12d 6708 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2925, 28eleqtrrd 2733 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑀 ∈ (𝑥𝐻𝑦))
30 funcco.n . . . . . . . . . 10 (𝜑𝑁 ∈ (𝑌𝐻𝑍))
3130ad4antr 769 . . . . . . . . 9 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑁 ∈ (𝑌𝐻𝑍))
32 simpllr 815 . . . . . . . . . 10 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑦 = 𝑌)
33 simplr 807 . . . . . . . . . 10 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑧 = 𝑍)
3432, 33oveq12d 6708 . . . . . . . . 9 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
3531, 34eleqtrrd 2733 . . . . . . . 8 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → 𝑁 ∈ (𝑦𝐻𝑧))
36 simp-5r 826 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑥 = 𝑋)
37 simpllr 815 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑧 = 𝑍)
3836, 37oveq12d 6708 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑥𝐺𝑧) = (𝑋𝐺𝑍))
39 simp-4r 824 . . . . . . . . . . . . 13 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑦 = 𝑌)
4036, 39opeq12d 4441 . . . . . . . . . . . 12 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
4140, 37oveq12d 6708 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑋, 𝑌· 𝑍))
42 simpr 476 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
43 simplr 807 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀)
4441, 42, 43oveq123d 6711 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑛(⟨𝑥, 𝑦· 𝑧)𝑚) = (𝑁(⟨𝑋, 𝑌· 𝑍)𝑀))
4538, 44fveq12d 6235 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)))
4636fveq2d 6233 . . . . . . . . . . . 12 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹𝑥) = (𝐹𝑋))
4739fveq2d 6233 . . . . . . . . . . . 12 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹𝑦) = (𝐹𝑌))
4846, 47opeq12d 4441 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑋), (𝐹𝑌)⟩)
4937fveq2d 6233 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝐹𝑧) = (𝐹𝑍))
5048, 49oveq12d 6708 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧)) = (⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍)))
5139, 37oveq12d 6708 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑦𝐺𝑧) = (𝑌𝐺𝑍))
5251, 42fveq12d 6235 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑦𝐺𝑧)‘𝑛) = ((𝑌𝐺𝑍)‘𝑁))
5336, 39oveq12d 6708 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (𝑥𝐺𝑦) = (𝑋𝐺𝑌))
5453, 43fveq12d 6235 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → ((𝑥𝐺𝑦)‘𝑚) = ((𝑋𝐺𝑌)‘𝑀))
5550, 52, 54oveq123d 6711 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀)))
5645, 55eqeq12d 2666 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) ∧ 𝑛 = 𝑁) → (((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) ↔ ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
5735, 56rspcdv 3343 . . . . . . 7 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑚 = 𝑀) → (∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
5829, 57rspcimdv 3341 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
5923, 58rspcimdv 3341 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
6021, 59rspcimdv 3341 . . . 4 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
6160adantld 482 . . 3 ((𝜑𝑥 = 𝑋) → ((((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
6219, 61rspcimdv 3341 . 2 (𝜑 → (∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))) → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀))))
6318, 62mpd 15 1 (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cop 4216   class class class wbr 4685   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373   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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  df-reu 2948  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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ixp 7951  df-func 16565
This theorem is referenced by:  funcsect  16579  funcoppc  16582  cofucl  16595  funcres  16603  fthsect  16632  fthmon  16634  catcisolem  16803  prfcl  16890  evlfcllem  16908  curf1cl  16915  curf2cl  16918  curfcl  16919  uncfcurf  16926  yonedalem4c  16964
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