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Theorem funcrngcsetcALT 42600
Description: Alternate proof of funcrngcsetc 42599, using cofuval2 16814 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 42598, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 17057. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 42599. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcrngcsetcALT.r 𝑅 = (RngCat‘𝑈)
funcrngcsetcALT.s 𝑆 = (SetCat‘𝑈)
funcrngcsetcALT.b 𝐵 = (Base‘𝑅)
funcrngcsetcALT.u (𝜑𝑈 ∈ WUni)
funcrngcsetcALT.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcrngcsetcALT.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
Assertion
Ref Expression
funcrngcsetcALT (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcrngcsetcALT
Dummy variables 𝑓 𝑔 𝑢 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcrngcsetcALT.f . . . . . . 7 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
2 fveq2 6375 . . . . . . . 8 (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
32cbvmptv 4909 . . . . . . 7 (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑢𝐵 ↦ (Base‘𝑢))
41, 3syl6eq 2815 . . . . . 6 (𝜑𝐹 = (𝑢𝐵 ↦ (Base‘𝑢)))
5 coires1 5839 . . . . . . 7 ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)) = ((𝑢𝑈 ↦ (Base‘𝑢)) ↾ 𝐵)
6 funcrngcsetcALT.r . . . . . . . . . . . 12 𝑅 = (RngCat‘𝑈)
7 funcrngcsetcALT.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑅)
8 funcrngcsetcALT.u . . . . . . . . . . . 12 (𝜑𝑈 ∈ WUni)
96, 7, 8rngcbas 42566 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Rng))
109eleq2d 2830 . . . . . . . . . 10 (𝜑 → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
11 elin 3958 . . . . . . . . . . 11 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1211simplbi 491 . . . . . . . . . 10 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥𝑈)
1310, 12syl6bi 244 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥𝑈))
1413ssrdv 3767 . . . . . . . 8 (𝜑𝐵𝑈)
1514resmptd 5629 . . . . . . 7 (𝜑 → ((𝑢𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢𝐵 ↦ (Base‘𝑢)))
165, 15syl5req 2812 . . . . . 6 (𝜑 → (𝑢𝐵 ↦ (Base‘𝑢)) = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
174, 16eqtrd 2799 . . . . 5 (𝜑𝐹 = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18 funcrngcsetcALT.g . . . . . . 7 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
19 coires1 5839 . . . . . . . . 9 (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦))
20 eqid 2765 . . . . . . . . . . . . 13 (Base‘𝑥) = (Base‘𝑥)
21 eqid 2765 . . . . . . . . . . . . 13 (Base‘𝑦) = (Base‘𝑦)
2220, 21rnghmf 42500 . . . . . . . . . . . 12 (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23 fvex 6388 . . . . . . . . . . . . . 14 (Base‘𝑦) ∈ V
24 fvex 6388 . . . . . . . . . . . . . 14 (Base‘𝑥) ∈ V
2523, 24pm3.2i 462 . . . . . . . . . . . . 13 ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26 elmapg 8073 . . . . . . . . . . . . 13 (((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V) → (𝑧 ∈ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ↔ 𝑧:(Base‘𝑥)⟶(Base‘𝑦)))
2725, 26mp1i 13 . . . . . . . . . . . 12 ((𝜑𝑥𝐵𝑦𝐵) → (𝑧 ∈ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ↔ 𝑧:(Base‘𝑥)⟶(Base‘𝑦)))
2822, 27syl5ibr 237 . . . . . . . . . . 11 ((𝜑𝑥𝐵𝑦𝐵) → (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
2928ssrdv 3767 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 RngHomo 𝑦) ⊆ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
3029resabs1d 5603 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥 RngHomo 𝑦)))
3119, 30syl5req 2812 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ( I ↾ (𝑥 RngHomo 𝑦)) = (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))))
3231mpt2eq3dva 6917 . . . . . . 7 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
3318, 32eqtrd 2799 . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
347a1i 11 . . . . . . 7 (𝜑𝐵 = (Base‘𝑅))
357a1i 11 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐵 = (Base‘𝑅))
36 fvresi 6632 . . . . . . . . . . . 12 (𝑥𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3736adantr 472 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3837adantl 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39 fvresi 6632 . . . . . . . . . . . 12 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4039adantl 473 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4140adantl 473 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4238, 41oveq12d 6860 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) = (𝑥(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))𝑦))
43 eqidd 2766 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) = (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))))
44 simprr 789 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑧 = 𝑦)
4544fveq2d 6379 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46 simprl 787 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑤 = 𝑥)
4746fveq2d 6379 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
4845, 47oveq12d 6860 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ((Base‘𝑧) ↑𝑚 (Base‘𝑤)) = ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
4948reseq2d 5565 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))) = ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
5013com12 32 . . . . . . . . . . . 12 (𝑥𝐵 → (𝜑𝑥𝑈))
5150adantr 472 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝜑𝑥𝑈))
5251impcom 396 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝑈)
539eleq2d 2830 . . . . . . . . . . . . 13 (𝜑 → (𝑦𝐵𝑦 ∈ (𝑈 ∩ Rng)))
54 elin 3958 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦𝑈𝑦 ∈ Rng))
5554simplbi 491 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦𝑈)
5653, 55syl6bi 244 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐵𝑦𝑈))
5756a1d 25 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → (𝑦𝐵𝑦𝑈)))
5857imp32 409 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝑈)
59 ovex 6874 . . . . . . . . . . . 12 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
6059a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V)
6160resiexd 6673 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
6243, 49, 52, 58, 61ovmpt2d 6986 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))𝑦) = ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
6342, 62eqtr2d 2800 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)))
64 eqidd 2766 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
65 oveq12 6851 . . . . . . . . . . . 12 ((𝑓 = 𝑥𝑔 = 𝑦) → (𝑓 RngHomo 𝑔) = (𝑥 RngHomo 𝑦))
6665reseq2d 5565 . . . . . . . . . . 11 ((𝑓 = 𝑥𝑔 = 𝑦) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
6766adantl 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑓 = 𝑥𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
68 simprl 787 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
69 simprr 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
70 ovex 6874 . . . . . . . . . . . 12 (𝑥 RngHomo 𝑦) ∈ V
7170a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) ∈ V)
7271resiexd 6673 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) ∈ V)
7364, 67, 68, 69, 72ovmpt2d 6986 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦) = ( I ↾ (𝑥 RngHomo 𝑦)))
7473eqcomd 2771 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))
7563, 74coeq12d 5455 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦)))
7634, 35, 75mpt2eq123dva 6914 . . . . . 6 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))))
7733, 76eqtrd 2799 . . . . 5 (𝜑𝐺 = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))))
7817, 77opeq12d 4567 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦)))⟩)
79 eqid 2765 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
80 eqid 2765 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81 eqidd 2766 . . . . . 6 (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82 eqidd 2766 . . . . . 6 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
836, 80, 7, 8, 81, 82rngcifuestrc 42598 . . . . 5 (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
84 funcrngcsetcALT.s . . . . . 6 𝑆 = (SetCat‘𝑈)
85 eqid 2765 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86 eqid 2765 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
8780, 8estrcbas 17032 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
8887mpteq1d 4897 . . . . . 6 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89 fveq2 6375 . . . . . . . . . . 11 (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
9089oveq2d 6858 . . . . . . . . . 10 (𝑤 = 𝑢 → ((Base‘𝑧) ↑𝑚 (Base‘𝑤)) = ((Base‘𝑧) ↑𝑚 (Base‘𝑢)))
9190reseq2d 5565 . . . . . . . . 9 (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑢))))
92 fveq2 6375 . . . . . . . . . . 11 (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
9392oveq1d 6857 . . . . . . . . . 10 (𝑧 = 𝑣 → ((Base‘𝑧) ↑𝑚 (Base‘𝑢)) = ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))
9493reseq2d 5565 . . . . . . . . 9 (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))))
9591, 94cbvmpt2v 6933 . . . . . . . 8 (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) = (𝑢𝑈, 𝑣𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))))
9695a1i 11 . . . . . . 7 (𝜑 → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) = (𝑢𝑈, 𝑣𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))))
97 eqidd 2766 . . . . . . . 8 (𝜑 → ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))))
9887, 87, 97mpt2eq123dv 6915 . . . . . . 7 (𝜑 → (𝑢𝑈, 𝑣𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑣 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))))
9996, 98eqtrd 2799 . . . . . 6 (𝜑 → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑣 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))))
10080, 84, 85, 86, 8, 88, 99funcestrcsetc 17057 . . . . 5 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))))
10179, 83, 100cofuval2 16814 . . . 4 (𝜑 → (⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩) = ⟨((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦)))⟩)
10278, 101eqtr4d 2802 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩))
103 df-br 4810 . . . . 5 (( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) ↔ ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
10483, 103sylib 209 . . . 4 (𝜑 → ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105 df-br 4810 . . . . 5 ((𝑢𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) ↔ ⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
106100, 105sylib 209 . . . 4 (𝜑 → ⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107104, 106cofucl 16815 . . 3 (𝜑 → (⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩) ∈ (𝑅 Func 𝑆))
108102, 107eqeltrd 2844 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109 df-br 4810 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110108, 109sylibr 225 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  cop 4340   class class class wbr 4809  cmpt 4888   I cid 5184  cres 5279  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑚 cmap 8060  WUnicwun 9775  Basecbs 16132   Func cfunc 16781  func ccofu 16783  SetCatcsetc 16992  ExtStrCatcestrc 17029  Rngcrng 42475   RngHomo crngh 42486  RngCatcrngc 42558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-wun 9777  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-hom 16240  df-cco 16241  df-0g 16370  df-cat 16596  df-cid 16597  df-homf 16598  df-ssc 16737  df-resc 16738  df-subc 16739  df-func 16785  df-idfu 16786  df-cofu 16787  df-full 16831  df-fth 16832  df-setc 16993  df-estrc 17030  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-mhm 17603  df-grp 17694  df-ghm 17924  df-abl 18462  df-mgp 18757  df-mgmhm 42380  df-rng0 42476  df-rnghomo 42488  df-rngc 42560
This theorem is referenced by: (None)
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