Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcrngcsetcALT Structured version   Visualization version   GIF version

Theorem funcrngcsetcALT 41791
 Description: Alternate proof of funcrngcsetc 41790, using cofuval2 16370 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 41789, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16612. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 41790. (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 6103 . . . . . . . 8 (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
32cbvmptv 4678 . . . . . . 7 (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑢𝐵 ↦ (Base‘𝑢))
41, 3syl6eq 2660 . . . . . 6 (𝜑𝐹 = (𝑢𝐵 ↦ (Base‘𝑢)))
5 coires1 5570 . . . . . . 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 41757 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Rng))
109eleq2d 2673 . . . . . . . . . 10 (𝜑 → (𝑥𝐵𝑥 ∈ (𝑈 ∩ Rng)))
11 elin 3758 . . . . . . . . . . 11 (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥𝑈𝑥 ∈ Rng))
1211simplbi 475 . . . . . . . . . 10 (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥𝑈)
1310, 12syl6bi 242 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥𝑈))
1413ssrdv 3574 . . . . . . . 8 (𝜑𝐵𝑈)
1514resmptd 5371 . . . . . . 7 (𝜑 → ((𝑢𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢𝐵 ↦ (Base‘𝑢)))
165, 15syl5req 2657 . . . . . 6 (𝜑 → (𝑢𝐵 ↦ (Base‘𝑢)) = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
174, 16eqtrd 2644 . . . . 5 (𝜑𝐹 = ((𝑢𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18 funcrngcsetcALT.g . . . . . . 7 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
19 coires1 5570 . . . . . . . . 9 (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦))
20 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑥) = (Base‘𝑥)
21 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑦) = (Base‘𝑦)
2220, 21rnghmf 41689 . . . . . . . . . . . 12 (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23 fvex 6113 . . . . . . . . . . . . . 14 (Base‘𝑦) ∈ V
24 fvex 6113 . . . . . . . . . . . . . 14 (Base‘𝑥) ∈ V
2523, 24pm3.2i 470 . . . . . . . . . . . . 13 ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26 elmapg 7757 . . . . . . . . . . . . 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 235 . . . . . . . . . . 11 ((𝜑𝑥𝐵𝑦𝐵) → (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
2928ssrdv 3574 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 RngHomo 𝑦) ⊆ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
3029resabs1d 5348 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥 RngHomo 𝑦)))
3119, 30syl5req 2657 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ( I ↾ (𝑥 RngHomo 𝑦)) = (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))))
3231mpt2eq3dva 6617 . . . . . . 7 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
3318, 32eqtrd 2644 . . . . . 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 6344 . . . . . . . . . . . 12 (𝑥𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3736adantr 480 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
3837adantl 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39 fvresi 6344 . . . . . . . . . . . 12 (𝑦𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4039adantl 481 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4140adantl 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
4238, 41oveq12d 6567 . . . . . . . . 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 2611 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))) = (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))))
44 simprr 792 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑧 = 𝑦)
4544fveq2d 6107 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46 simprl 790 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → 𝑤 = 𝑥)
4746fveq2d 6107 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
4845, 47oveq12d 6567 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ((Base‘𝑧) ↑𝑚 (Base‘𝑤)) = ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
4948reseq2d 5317 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑤 = 𝑥𝑧 = 𝑦)) → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))) = ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
5013com12 32 . . . . . . . . . . . 12 (𝑥𝐵 → (𝜑𝑥𝑈))
5150adantr 480 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵) → (𝜑𝑥𝑈))
5251impcom 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝑈)
539eleq2d 2673 . . . . . . . . . . . . 13 (𝜑 → (𝑦𝐵𝑦 ∈ (𝑈 ∩ Rng)))
54 elin 3758 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦𝑈𝑦 ∈ Rng))
5554simplbi 475 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦𝑈)
5653, 55syl6bi 242 . . . . . . . . . . . 12 (𝜑 → (𝑦𝐵𝑦𝑈))
5756a1d 25 . . . . . . . . . . 11 (𝜑 → (𝑥𝐵 → (𝑦𝐵𝑦𝑈)))
5857imp32 448 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝑈)
59 ovex 6577 . . . . . . . . . . . 12 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
6059a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V)
6160resiexd 6385 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
6243, 49, 52, 58, 61ovmpt2d 6686 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))𝑦) = ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
6342, 62eqtr2d 2645 . . . . . . . 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 2611 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
65 oveq12 6558 . . . . . . . . . . . 12 ((𝑓 = 𝑥𝑔 = 𝑦) → (𝑓 RngHomo 𝑔) = (𝑥 RngHomo 𝑦))
6665reseq2d 5317 . . . . . . . . . . 11 ((𝑓 = 𝑥𝑔 = 𝑦) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
6766adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ (𝑓 = 𝑥𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
68 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
69 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
70 ovex 6577 . . . . . . . . . . . 12 (𝑥 RngHomo 𝑦) ∈ V
7170a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) ∈ V)
7271resiexd 6385 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) ∈ V)
7364, 67, 68, 69, 72ovmpt2d 6686 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦) = ( I ↾ (𝑥 RngHomo 𝑦)))
7473eqcomd 2616 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = (𝑥(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))
7563, 74coeq12d 5208 . . . . . . 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 6614 . . . . . 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 2644 . . . . 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 4348 . . . 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 2610 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
80 eqid 2610 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81 eqidd 2611 . . . . . 6 (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82 eqidd 2611 . . . . . 6 (𝜑 → (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
836, 80, 7, 8, 81, 82rngcifuestrc 41789 . . . . 5 (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
84 funcrngcsetcALT.s . . . . . 6 𝑆 = (SetCat‘𝑈)
85 eqid 2610 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86 eqid 2610 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
8780, 8estrcbas 16588 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
8887mpteq1d 4666 . . . . . 6 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89 fveq2 6103 . . . . . . . . . . 11 (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
9089oveq2d 6565 . . . . . . . . . 10 (𝑤 = 𝑢 → ((Base‘𝑧) ↑𝑚 (Base‘𝑤)) = ((Base‘𝑧) ↑𝑚 (Base‘𝑢)))
9190reseq2d 5317 . . . . . . . . 9 (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑢))))
92 fveq2 6103 . . . . . . . . . . 11 (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
9392oveq1d 6564 . . . . . . . . . 10 (𝑧 = 𝑣 → ((Base‘𝑧) ↑𝑚 (Base‘𝑢)) = ((Base‘𝑣) ↑𝑚 (Base‘𝑢)))
9493reseq2d 5317 . . . . . . . . 9 (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))))
9591, 94cbvmpt2v 6633 . . . . . . . 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 2611 . . . . . . . 8 (𝜑 → ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑𝑚 (Base‘𝑢))))
9887, 87, 97mpt2eq123dv 6615 . . . . . . 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 2644 . . . . . 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 16612 . . . . 5 (𝜑 → (𝑢𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤)))))
10179, 83, 100cofuval2 16370 . . . 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 2647 . . 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 4584 . . . . 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 207 . . . 4 (𝜑 → ⟨( I ↾ 𝐵), (𝑓𝐵, 𝑔𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105 df-br 4584 . . . . 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 207 . . . 4 (𝜑 → ⟨(𝑢𝑈 ↦ (Base‘𝑢)), (𝑤𝑈, 𝑧𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑𝑚 (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107104, 106cofucl 16371 . . 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 2688 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109 df-br 4584 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110108, 109sylibr 223 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  WUnicwun 9401  Basecbs 15695   Func cfunc 16337   ∘func ccofu 16339  SetCatcsetc 16548  ExtStrCatcestrc 16585  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295  df-func 16341  df-idfu 16342  df-cofu 16343  df-full 16387  df-fth 16388  df-setc 16549  df-estrc 16586  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-abl 18019  df-mgp 18313  df-mgmhm 41569  df-rng0 41665  df-rnghomo 41677  df-rngc 41751 This theorem is referenced by: (None)
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