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Theorem funcrngcsetcALT 46287
Description: Alternate proof of funcrngcsetc 46286, using cofuval2 17773 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 46285, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18037. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 46286. (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 6842 . . . . . . . 8 (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
32cbvmptv 5218 . . . . . . 7 (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑢𝐵 ↦ (Base‘𝑢))
41, 3eqtrdi 2792 . . . . . 6 (𝜑𝐹 = (𝑢𝐵 ↦ (Base‘𝑢)))
5 coires1 6216 . . . . . . 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 46253 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Rng))
109eleq2d 2823 . . . . . . . . . 10 (𝜑 → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
11 elin 3926 . . . . . . . . . . 11 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1211simplbi 498 . . . . . . . . . 10 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥𝑈)
1310, 12syl6bi 252 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥𝑈))
1413ssrdv 3950 . . . . . . . 8 (𝜑𝐵𝑈)
1514resmptd 5994 . . . . . . 7 (𝜑 → ((𝑢𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢𝐵 ↦ (Base‘𝑢)))
165, 15eqtr2id 2789 . . . . . 6 (𝜑 → (𝑢𝐵 ↦ (Base‘𝑢)) = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
174, 16eqtrd 2776 . . . . 5 (𝜑𝐹 = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18 funcrngcsetcALT.g . . . . . . 7 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
19 coires1 6216 . . . . . . . . 9 (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦))
20 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝑥) = (Base‘𝑥)
21 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝑦) = (Base‘𝑦)
2220, 21rnghmf 46187 . . . . . . . . . . . 12 (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23 fvex 6855 . . . . . . . . . . . . . 14 (Base‘𝑦) ∈ V
24 fvex 6855 . . . . . . . . . . . . . 14 (Base‘𝑥) ∈ V
2523, 24pm3.2i 471 . . . . . . . . . . . . 13 ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26 elmapg 8778 . . . . . . . . . . . . 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 245 . . . . . . . . . . 11 ((𝜑𝑥𝐵𝑦𝐵) → (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥))))
2928ssrdv 3950 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 RngHomo 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
3029resabs1d 5968 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥 RngHomo 𝑦)))
3119, 30eqtr2id 2789 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ( I ↾ (𝑥 RngHomo 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))))
3231mpoeq3dva 7434 . . . . . . 7 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
3318, 32eqtrd 2776 . . . . . 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 7119 . . . . . . . . . . . 12 (𝑥𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3736adantr 481 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3837adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39 fvresi 7119 . . . . . . . . . . . 12 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4039adantl 482 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4140adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4238, 41oveq12d 7375 . . . . . . . . 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 2737 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
44 simprr 771 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑧 = 𝑦)
4544fveq2d 6846 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46 simprl 769 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑤 = 𝑥)
4746fveq2d 6846 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
4845, 47oveq12d 7375 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
4948reseq2d 5937 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
5013com12 32 . . . . . . . . . . . 12 (𝑥𝐵 → (𝜑𝑥𝑈))
5150adantr 481 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝜑𝑥𝑈))
5251impcom 408 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝑈)
539eleq2d 2823 . . . . . . . . . . . . 13 (𝜑 → (𝑦𝐵𝑦 ∈ (𝑈 ∩ Rng)))
54 elin 3926 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦𝑈𝑦 ∈ Rng))
5554simplbi 498 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦𝑈)
5653, 55syl6bi 252 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐵𝑦𝑈))
5756a1d 25 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → (𝑦𝐵𝑦𝑈)))
5857imp32 419 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝑈)
59 ovex 7390 . . . . . . . . . . . 12 ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
6059a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V)
6160resiexd 7166 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
6243, 49, 52, 58, 61ovmpod 7507 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))𝑦) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
6342, 62eqtr2d 2777 . . . . . . . 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 2737 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
65 oveq12 7366 . . . . . . . . . . . 12 ((𝑓 = 𝑥𝑔 = 𝑦) → (𝑓 RngHomo 𝑔) = (𝑥 RngHomo 𝑦))
6665reseq2d 5937 . . . . . . . . . . 11 ((𝑓 = 𝑥𝑔 = 𝑦) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
6766adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑓 = 𝑥𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
68 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
69 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
70 ovex 7390 . . . . . . . . . . . 12 (𝑥 RngHomo 𝑦) ∈ V
7170a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) ∈ V)
7271resiexd 7166 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) ∈ V)
7364, 67, 68, 69, 72ovmpod 7507 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦) = ( I ↾ (𝑥 RngHomo 𝑦)))
7473eqcomd 2742 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))
7563, 74coeq12d 5820 . . . . . . 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 7431 . . . . . 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 2776 . . . . 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 4838 . . . 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 2736 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
80 eqid 2736 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81 eqidd 2737 . . . . . 6 (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82 eqidd 2737 . . . . . 6 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
836, 80, 7, 8, 81, 82rngcifuestrc 46285 . . . . 5 (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
84 funcrngcsetcALT.s . . . . . 6 𝑆 = (SetCat‘𝑈)
85 eqid 2736 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86 eqid 2736 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
8780, 8estrcbas 18012 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
8887mpteq1d 5200 . . . . . 6 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89 fveq2 6842 . . . . . . . . . . 11 (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
9089oveq2d 7373 . . . . . . . . . 10 (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
9190reseq2d 5937 . . . . . . . . 9 (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92 fveq2 6842 . . . . . . . . . . 11 (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
9392oveq1d 7372 . . . . . . . . . 10 (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
9493reseq2d 5937 . . . . . . . . 9 (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
9591, 94cbvmpov 7452 . . . . . . . 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 2737 . . . . . . . 8 (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
9887, 87, 97mpoeq123dv 7432 . . . . . . 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 2776 . . . . . 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 18037 . . . . 5 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
10179, 83, 100cofuval2 17773 . . . 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 2779 . . 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 5106 . . . . 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 217 . . . 4 (𝜑 → ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105 df-br 5106 . . . . 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 217 . . . 4 (𝜑 → ⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107104, 106cofucl 17774 . . 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 2838 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109 df-br 5106 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110108, 109sylibr 233 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cin 3909  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530  cres 5635  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  WUnicwun 10636  Basecbs 17083   Func cfunc 17740  func ccofu 17742  SetCatcsetc 17961  ExtStrCatcestrc 18009  Rngcrng 46162   RngHomo crngh 46173  RngCatcrngc 46245
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wun 10638  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-hom 17157  df-cco 17158  df-0g 17323  df-cat 17548  df-cid 17549  df-homf 17550  df-ssc 17693  df-resc 17694  df-subc 17695  df-func 17744  df-idfu 17745  df-cofu 17746  df-full 17791  df-fth 17792  df-setc 17962  df-estrc 18010  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-abl 19565  df-mgp 19897  df-mgmhm 46063  df-rng 46163  df-rnghomo 46175  df-rngc 46247
This theorem is referenced by: (None)
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