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

Theorem iscatd 16157
Description: Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
iscatd.b (𝜑𝐵 = (Base‘𝐶))
iscatd.h (𝜑𝐻 = (Hom ‘𝐶))
iscatd.o (𝜑· = (comp‘𝐶))
iscatd.c (𝜑𝐶𝑉)
iscatd.1 ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))
iscatd.2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)
iscatd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)
iscatd.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
iscatd.5 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
Assertion
Ref Expression
iscatd (𝜑𝐶 ∈ Cat)
Distinct variable groups:   𝑓,𝑔,𝑦, 1   𝑓,𝑘,𝑤,𝑥,𝑧,𝐵,𝑔,𝑦   𝜑,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   · ,𝑔   𝐶,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   𝑓,𝐻,𝑔,𝑘,𝑤
Allowed substitution hints:   · (𝑥,𝑦,𝑧,𝑤,𝑓,𝑘)   1 (𝑥,𝑧,𝑤,𝑘)   𝐻(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)

Proof of Theorem iscatd
StepHypRef Expression
1 iscatd.1 . . . . . 6 ((𝜑𝑥𝐵) → 1 ∈ (𝑥𝐻𝑥))
2 iscatd.2 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)
323exp2 1277 . . . . . . . . . 10 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑓 ∈ (𝑦𝐻𝑥) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))))
43imp31 447 . . . . . . . . 9 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑓 ∈ (𝑦𝐻𝑥) → ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
54ralrimiv 2948 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓)
6 iscatd.3 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)
763exp2 1277 . . . . . . . . . 10 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑓 ∈ (𝑥𝐻𝑦) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))))
87imp31 447 . . . . . . . . 9 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (𝑓 ∈ (𝑥𝐻𝑦) → (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
98ralrimiv 2948 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)
105, 9jca 553 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
1110ralrimiva 2949 . . . . . 6 ((𝜑𝑥𝐵) → ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
12 oveq1 6556 . . . . . . . . . . 11 (𝑔 = 1 → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓))
1312eqeq1d 2612 . . . . . . . . . 10 (𝑔 = 1 → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
1413ralbidv 2969 . . . . . . . . 9 (𝑔 = 1 → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓))
15 oveq2 6557 . . . . . . . . . . 11 (𝑔 = 1 → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ))
1615eqeq1d 2612 . . . . . . . . . 10 (𝑔 = 1 → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
1716ralbidv 2969 . . . . . . . . 9 (𝑔 = 1 → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓))
1814, 17anbi12d 743 . . . . . . . 8 (𝑔 = 1 → ((∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)))
1918ralbidv 2969 . . . . . . 7 (𝑔 = 1 → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)))
2019rspcev 3282 . . . . . 6 (( 1 ∈ (𝑥𝐻𝑥) ∧ ∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)( 1 (⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦) 1 ) = 𝑓)) → ∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
211, 11, 20syl2anc 691 . . . . 5 ((𝜑𝑥𝐵) → ∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓))
22 iscatd.4 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
23223expia 1259 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
24233exp2 1277 . . . . . . . . 9 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))))))
2524imp43 619 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
26 iscatd.5 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
27263expa 1257 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵))) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
28273exp2 1277 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵))) → (𝑓 ∈ (𝑥𝐻𝑦) → (𝑔 ∈ (𝑦𝐻𝑧) → (𝑘 ∈ (𝑧𝐻𝑤) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))
2928imp32 448 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵))) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑘 ∈ (𝑧𝐻𝑤) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
3029ralrimiv 2948 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵))) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
3130ex 449 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵))) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
3231expr 641 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑧𝐵𝑤𝐵) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
3332expd 451 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑧𝐵 → (𝑤𝐵 → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))
3433expr 641 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑦𝐵 → (𝑧𝐵 → (𝑤𝐵 → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))))
3534imp42 618 . . . . . . . . 9 ((((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) ∧ 𝑤𝐵) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
3635ralrimdva 2952 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
3725, 36jcad 554 . . . . . . 7 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
3837ralrimivv 2953 . . . . . 6 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
3938ralrimivva 2954 . . . . 5 ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))
4021, 39jca 553 . . . 4 ((𝜑𝑥𝐵) → (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
4140ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
42 iscatd.b . . . 4 (𝜑𝐵 = (Base‘𝐶))
43 iscatd.h . . . . . . 7 (𝜑𝐻 = (Hom ‘𝐶))
4443oveqd 6566 . . . . . 6 (𝜑 → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
4543oveqd 6566 . . . . . . . . 9 (𝜑 → (𝑦𝐻𝑥) = (𝑦(Hom ‘𝐶)𝑥))
46 iscatd.o . . . . . . . . . . . 12 (𝜑· = (comp‘𝐶))
4746oveqd 6566 . . . . . . . . . . 11 (𝜑 → (⟨𝑦, 𝑥· 𝑥) = (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥))
4847oveqd 6566 . . . . . . . . . 10 (𝜑 → (𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
4948eqeq1d 2612 . . . . . . . . 9 (𝜑 → ((𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
5045, 49raleqbidv 3129 . . . . . . . 8 (𝜑 → (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
5143oveqd 6566 . . . . . . . . 9 (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
5246oveqd 6566 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑥· 𝑦) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦))
5352oveqd 6566 . . . . . . . . . 10 (𝜑 → (𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔))
5453eqeq1d 2612 . . . . . . . . 9 (𝜑 → ((𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
5551, 54raleqbidv 3129 . . . . . . . 8 (𝜑 → (∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
5650, 55anbi12d 743 . . . . . . 7 (𝜑 → ((∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
5742, 56raleqbidv 3129 . . . . . 6 (𝜑 → (∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
5844, 57rexeqbidv 3130 . . . . 5 (𝜑 → (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ↔ ∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)))
5943oveqd 6566 . . . . . . . . 9 (𝜑 → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
6046oveqd 6566 . . . . . . . . . . . 12 (𝜑 → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧))
6160oveqd 6566 . . . . . . . . . . 11 (𝜑 → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
6243oveqd 6566 . . . . . . . . . . 11 (𝜑 → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
6361, 62eleq12d 2682 . . . . . . . . . 10 (𝜑 → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
6443oveqd 6566 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐻𝑤) = (𝑧(Hom ‘𝐶)𝑤))
6546oveqd 6566 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑥, 𝑦· 𝑤) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤))
6646oveqd 6566 . . . . . . . . . . . . . . 15 (𝜑 → (⟨𝑦, 𝑧· 𝑤) = (⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤))
6766oveqd 6566 . . . . . . . . . . . . . 14 (𝜑 → (𝑘(⟨𝑦, 𝑧· 𝑤)𝑔) = (𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
68 eqidd 2611 . . . . . . . . . . . . . 14 (𝜑𝑓 = 𝑓)
6965, 67, 68oveq123d 6570 . . . . . . . . . . . . 13 (𝜑 → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = ((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓))
7046oveqd 6566 . . . . . . . . . . . . . 14 (𝜑 → (⟨𝑥, 𝑧· 𝑤) = (⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤))
71 eqidd 2611 . . . . . . . . . . . . . 14 (𝜑𝑘 = 𝑘)
7270, 71, 61oveq123d 6570 . . . . . . . . . . . . 13 (𝜑 → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
7369, 72eqeq12d 2625 . . . . . . . . . . . 12 (𝜑 → (((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) ↔ ((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
7464, 73raleqbidv 3129 . . . . . . . . . . 11 (𝜑 → (∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) ↔ ∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
7542, 74raleqbidv 3129 . . . . . . . . . 10 (𝜑 → (∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) ↔ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
7663, 75anbi12d 743 . . . . . . . . 9 (𝜑 → (((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
7759, 76raleqbidv 3129 . . . . . . . 8 (𝜑 → (∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
7851, 77raleqbidv 3129 . . . . . . 7 (𝜑 → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
7942, 78raleqbidv 3129 . . . . . 6 (𝜑 → (∀𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
8042, 79raleqbidv 3129 . . . . 5 (𝜑 → (∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
8158, 80anbi12d 743 . . . 4 (𝜑 → ((∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) ↔ (∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
8242, 81raleqbidv 3129 . . 3 (𝜑 → (∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) ↔ ∀𝑥 ∈ (Base‘𝐶)(∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
8341, 82mpbid 221 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)(∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
84 iscatd.c . . 3 (𝜑𝐶𝑉)
85 eqid 2610 . . . 4 (Base‘𝐶) = (Base‘𝐶)
86 eqid 2610 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
87 eqid 2610 . . . 4 (comp‘𝐶) = (comp‘𝐶)
8885, 86, 87iscat 16156 . . 3 (𝐶𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ (Base‘𝐶)(∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
8984, 88syl 17 . 2 (𝜑 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ (Base‘𝐶)(∃𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
9083, 89mpbird 246 1 (𝜑𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cop 4131  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148
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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cat 16152
This theorem is referenced by:  iscatd2  16165  0catg  16171
  Copyright terms: Public domain W3C validator