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Theorem funcrngcsetcALT 44415
Description: Alternate proof of funcrngcsetc 44414, using cofuval2 17135 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 44413, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 17377. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 44414. (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 6643 . . . . . . . 8 (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
32cbvmptv 5142 . . . . . . 7 (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑢𝐵 ↦ (Base‘𝑢))
41, 3syl6eq 2872 . . . . . 6 (𝜑𝐹 = (𝑢𝐵 ↦ (Base‘𝑢)))
5 coires1 6090 . . . . . . 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 44381 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Rng))
109eleq2d 2897 . . . . . . . . . 10 (𝜑 → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
11 elin 3926 . . . . . . . . . . 11 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1211simplbi 501 . . . . . . . . . 10 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥𝑈)
1310, 12syl6bi 256 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥𝑈))
1413ssrdv 3949 . . . . . . . 8 (𝜑𝐵𝑈)
1514resmptd 5881 . . . . . . 7 (𝜑 → ((𝑢𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢𝐵 ↦ (Base‘𝑢)))
165, 15syl5req 2869 . . . . . 6 (𝜑 → (𝑢𝐵 ↦ (Base‘𝑢)) = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
174, 16eqtrd 2856 . . . . 5 (𝜑𝐹 = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18 funcrngcsetcALT.g . . . . . . 7 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
19 coires1 6090 . . . . . . . . 9 (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦))
20 eqid 2821 . . . . . . . . . . . . 13 (Base‘𝑥) = (Base‘𝑥)
21 eqid 2821 . . . . . . . . . . . . 13 (Base‘𝑦) = (Base‘𝑦)
2220, 21rnghmf 44315 . . . . . . . . . . . 12 (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23 fvex 6656 . . . . . . . . . . . . . 14 (Base‘𝑦) ∈ V
24 fvex 6656 . . . . . . . . . . . . . 14 (Base‘𝑥) ∈ V
2523, 24pm3.2i 474 . . . . . . . . . . . . 13 ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26 elmapg 8394 . . . . . . . . . . . . 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 249 . . . . . . . . . . 11 ((𝜑𝑥𝐵𝑦𝐵) → (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥))))
2928ssrdv 3949 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 RngHomo 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
3029resabs1d 5857 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥 RngHomo 𝑦)))
3119, 30syl5req 2869 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ( I ↾ (𝑥 RngHomo 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))))
3231mpoeq3dva 7205 . . . . . . 7 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
3318, 32eqtrd 2856 . . . . . 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 6908 . . . . . . . . . . . 12 (𝑥𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3736adantr 484 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3837adantl 485 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39 fvresi 6908 . . . . . . . . . . . 12 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4039adantl 485 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4140adantl 485 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4238, 41oveq12d 7148 . . . . . . . . 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 2822 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
44 simprr 772 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑧 = 𝑦)
4544fveq2d 6647 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46 simprl 770 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑤 = 𝑥)
4746fveq2d 6647 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
4845, 47oveq12d 7148 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
4948reseq2d 5826 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
5013com12 32 . . . . . . . . . . . 12 (𝑥𝐵 → (𝜑𝑥𝑈))
5150adantr 484 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝜑𝑥𝑈))
5251impcom 411 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝑈)
539eleq2d 2897 . . . . . . . . . . . . 13 (𝜑 → (𝑦𝐵𝑦 ∈ (𝑈 ∩ Rng)))
54 elin 3926 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦𝑈𝑦 ∈ Rng))
5554simplbi 501 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦𝑈)
5653, 55syl6bi 256 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐵𝑦𝑈))
5756a1d 25 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → (𝑦𝐵𝑦𝑈)))
5857imp32 422 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝑈)
59 ovex 7163 . . . . . . . . . . . 12 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
6059a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V)
6160resiexd 6952 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
6243, 49, 52, 58, 61ovmpod 7276 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))𝑦) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
6342, 62eqtr2d 2857 . . . . . . . 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 2822 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
65 oveq12 7139 . . . . . . . . . . . 12 ((𝑓 = 𝑥𝑔 = 𝑦) → (𝑓 RngHomo 𝑔) = (𝑥 RngHomo 𝑦))
6665reseq2d 5826 . . . . . . . . . . 11 ((𝑓 = 𝑥𝑔 = 𝑦) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
6766adantl 485 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑓 = 𝑥𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
68 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
69 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
70 ovex 7163 . . . . . . . . . . . 12 (𝑥 RngHomo 𝑦) ∈ V
7170a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) ∈ V)
7271resiexd 6952 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) ∈ V)
7364, 67, 68, 69, 72ovmpod 7276 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦) = ( I ↾ (𝑥 RngHomo 𝑦)))
7473eqcomd 2827 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))
7563, 74coeq12d 5708 . . . . . . 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, 75mpoeq123dva 7202 . . . . . 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 2856 . . . . 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 4784 . . . 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 2821 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
80 eqid 2821 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81 eqidd 2822 . . . . . 6 (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82 eqidd 2822 . . . . . 6 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
836, 80, 7, 8, 81, 82rngcifuestrc 44413 . . . . 5 (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
84 funcrngcsetcALT.s . . . . . 6 𝑆 = (SetCat‘𝑈)
85 eqid 2821 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86 eqid 2821 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
8780, 8estrcbas 17353 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
8887mpteq1d 5128 . . . . . 6 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89 fveq2 6643 . . . . . . . . . . 11 (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
9089oveq2d 7146 . . . . . . . . . 10 (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
9190reseq2d 5826 . . . . . . . . 9 (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92 fveq2 6643 . . . . . . . . . . 11 (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
9392oveq1d 7145 . . . . . . . . . 10 (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
9493reseq2d 5826 . . . . . . . . 9 (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
9591, 94cbvmpov 7223 . . . . . . . 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 2822 . . . . . . . 8 (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
9887, 87, 97mpoeq123dv 7203 . . . . . . 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 2856 . . . . . 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 17377 . . . . 5 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
10179, 83, 100cofuval2 17135 . . . 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 2859 . . 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 5040 . . . . 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 221 . . . 4 (𝜑 → ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105 df-br 5040 . . . . 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 221 . . . 4 (𝜑 → ⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107104, 106cofucl 17136 . . 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 2912 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109 df-br 5040 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110108, 109sylibr 237 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3471  cin 3909  cop 4546   class class class wbr 5039  cmpt 5119   I cid 5432  cres 5530  ccom 5532  wf 6324  cfv 6328  (class class class)co 7130  cmpo 7132  m cmap 8381  WUnicwun 10099  Basecbs 16461   Func cfunc 17102  func ccofu 17104  SetCatcsetc 17313  ExtStrCatcestrc 17350  Rngcrng 44290   RngHomo crngh 44301  RngCatcrngc 44373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-wun 10101  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-fz 12876  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-hom 16567  df-cco 16568  df-0g 16693  df-cat 16917  df-cid 16918  df-homf 16919  df-ssc 17058  df-resc 17059  df-subc 17060  df-func 17106  df-idfu 17107  df-cofu 17108  df-full 17152  df-fth 17153  df-setc 17314  df-estrc 17351  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-mhm 17934  df-grp 18084  df-ghm 18334  df-abl 18887  df-mgp 19218  df-mgmhm 44191  df-rng0 44291  df-rnghomo 44303  df-rngc 44375
This theorem is referenced by: (None)
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