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Theorem funcrngcsetcALT 46229
Description: Alternate proof of funcrngcsetc 46228, using cofuval2 17765 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 46227, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18029. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 46228. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcrngcsetcALT.r 𝑅 = (RngCatβ€˜π‘ˆ)
funcrngcsetcALT.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcrngcsetcALT.b 𝐡 = (Baseβ€˜π‘…)
funcrngcsetcALT.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcrngcsetcALT.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcrngcsetcALT.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHomo 𝑦))))
Assertion
Ref Expression
funcrngcsetcALT (πœ‘ β†’ 𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝑅,𝑦   π‘₯,π‘ˆ,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcrngcsetcALT
Dummy variables 𝑓 𝑔 𝑒 𝑀 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcrngcsetcALT.f . . . . . . 7 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
2 fveq2 6839 . . . . . . . 8 (π‘₯ = 𝑒 β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘’))
32cbvmptv 5216 . . . . . . 7 (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)) = (𝑒 ∈ 𝐡 ↦ (Baseβ€˜π‘’))
41, 3eqtrdi 2792 . . . . . 6 (πœ‘ β†’ 𝐹 = (𝑒 ∈ 𝐡 ↦ (Baseβ€˜π‘’)))
5 coires1 6214 . . . . . . 7 ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)) = ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) β†Ύ 𝐡)
6 funcrngcsetcALT.r . . . . . . . . . . . 12 𝑅 = (RngCatβ€˜π‘ˆ)
7 funcrngcsetcALT.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜π‘…)
8 funcrngcsetcALT.u . . . . . . . . . . . 12 (πœ‘ β†’ π‘ˆ ∈ WUni)
96, 7, 8rngcbas 46195 . . . . . . . . . . 11 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Rng))
109eleq2d 2823 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↔ π‘₯ ∈ (π‘ˆ ∩ Rng)))
11 elin 3924 . . . . . . . . . . 11 (π‘₯ ∈ (π‘ˆ ∩ Rng) ↔ (π‘₯ ∈ π‘ˆ ∧ π‘₯ ∈ Rng))
1211simplbi 498 . . . . . . . . . 10 (π‘₯ ∈ (π‘ˆ ∩ Rng) β†’ π‘₯ ∈ π‘ˆ)
1310, 12syl6bi 252 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝐡 β†’ π‘₯ ∈ π‘ˆ))
1413ssrdv 3948 . . . . . . . 8 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
1514resmptd 5992 . . . . . . 7 (πœ‘ β†’ ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) β†Ύ 𝐡) = (𝑒 ∈ 𝐡 ↦ (Baseβ€˜π‘’)))
165, 15eqtr2id 2789 . . . . . 6 (πœ‘ β†’ (𝑒 ∈ 𝐡 ↦ (Baseβ€˜π‘’)) = ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)))
174, 16eqtrd 2776 . . . . 5 (πœ‘ β†’ 𝐹 = ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)))
18 funcrngcsetcALT.g . . . . . . 7 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHomo 𝑦))))
19 coires1 6214 . . . . . . . . 9 (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦))) = (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) β†Ύ (π‘₯ RngHomo 𝑦))
20 eqid 2736 . . . . . . . . . . . . 13 (Baseβ€˜π‘₯) = (Baseβ€˜π‘₯)
21 eqid 2736 . . . . . . . . . . . . 13 (Baseβ€˜π‘¦) = (Baseβ€˜π‘¦)
2220, 21rnghmf 46129 . . . . . . . . . . . 12 (𝑧 ∈ (π‘₯ RngHomo 𝑦) β†’ 𝑧:(Baseβ€˜π‘₯)⟢(Baseβ€˜π‘¦))
23 fvex 6852 . . . . . . . . . . . . . 14 (Baseβ€˜π‘¦) ∈ V
24 fvex 6852 . . . . . . . . . . . . . 14 (Baseβ€˜π‘₯) ∈ V
2523, 24pm3.2i 471 . . . . . . . . . . . . 13 ((Baseβ€˜π‘¦) ∈ V ∧ (Baseβ€˜π‘₯) ∈ V)
26 elmapg 8774 . . . . . . . . . . . . 13 (((Baseβ€˜π‘¦) ∈ V ∧ (Baseβ€˜π‘₯) ∈ V) β†’ (𝑧 ∈ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ↔ 𝑧:(Baseβ€˜π‘₯)⟢(Baseβ€˜π‘¦)))
2725, 26mp1i 13 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑧 ∈ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ↔ 𝑧:(Baseβ€˜π‘₯)⟢(Baseβ€˜π‘¦)))
2822, 27syl5ibr 245 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑧 ∈ (π‘₯ RngHomo 𝑦) β†’ 𝑧 ∈ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
2928ssrdv 3948 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ RngHomo 𝑦) βŠ† ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))
3029resabs1d 5966 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) β†Ύ (π‘₯ RngHomo 𝑦)) = ( I β†Ύ (π‘₯ RngHomo 𝑦)))
3119, 30eqtr2id 2789 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ( I β†Ύ (π‘₯ RngHomo 𝑦)) = (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦))))
3231mpoeq3dva 7430 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHomo 𝑦))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦)))))
3318, 32eqtrd 2776 . . . . . 6 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦)))))
347a1i 11 . . . . . . 7 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
357a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘…))
36 fvresi 7115 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
3736adantr 481 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
3837adantl 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
39 fvresi 7115 . . . . . . . . . . . 12 (𝑦 ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘¦) = 𝑦)
4039adantl 482 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (( I β†Ύ 𝐡)β€˜π‘¦) = 𝑦)
4140adantl 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ 𝐡)β€˜π‘¦) = 𝑦)
4238, 41oveq12d 7371 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) = (π‘₯(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))𝑦))
43 eqidd 2737 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))) = (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))))
44 simprr 771 . . . . . . . . . . . . 13 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ 𝑧 = 𝑦)
4544fveq2d 6843 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ (Baseβ€˜π‘§) = (Baseβ€˜π‘¦))
46 simprl 769 . . . . . . . . . . . . 13 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ 𝑀 = π‘₯)
4746fveq2d 6843 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘₯))
4845, 47oveq12d 7371 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)) = ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))
4948reseq2d 5935 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑀 = π‘₯ ∧ 𝑧 = 𝑦)) β†’ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))) = ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
5013com12 32 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐡 β†’ (πœ‘ β†’ π‘₯ ∈ π‘ˆ))
5150adantr 481 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (πœ‘ β†’ π‘₯ ∈ π‘ˆ))
5251impcom 408 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ π‘₯ ∈ π‘ˆ)
539eleq2d 2823 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↔ 𝑦 ∈ (π‘ˆ ∩ Rng)))
54 elin 3924 . . . . . . . . . . . . . 14 (𝑦 ∈ (π‘ˆ ∩ Rng) ↔ (𝑦 ∈ π‘ˆ ∧ 𝑦 ∈ Rng))
5554simplbi 498 . . . . . . . . . . . . 13 (𝑦 ∈ (π‘ˆ ∩ Rng) β†’ 𝑦 ∈ π‘ˆ)
5653, 55syl6bi 252 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ π‘ˆ))
5756a1d 25 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ 𝐡 β†’ (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ π‘ˆ)))
5857imp32 419 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝑦 ∈ π‘ˆ)
59 ovex 7386 . . . . . . . . . . . 12 ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ∈ V
6059a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ∈ V)
6160resiexd 7162 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∈ V)
6243, 49, 52, 58, 61ovmpod 7503 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))𝑦) = ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
6342, 62eqtr2d 2777 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)))
64 eqidd 2737 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))))
65 oveq12 7362 . . . . . . . . . . . 12 ((𝑓 = π‘₯ ∧ 𝑔 = 𝑦) β†’ (𝑓 RngHomo 𝑔) = (π‘₯ RngHomo 𝑦))
6665reseq2d 5935 . . . . . . . . . . 11 ((𝑓 = π‘₯ ∧ 𝑔 = 𝑦) β†’ ( I β†Ύ (𝑓 RngHomo 𝑔)) = ( I β†Ύ (π‘₯ RngHomo 𝑦)))
6766adantl 482 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) ∧ (𝑓 = π‘₯ ∧ 𝑔 = 𝑦)) β†’ ( I β†Ύ (𝑓 RngHomo 𝑔)) = ( I β†Ύ (π‘₯ RngHomo 𝑦)))
68 simprl 769 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ π‘₯ ∈ 𝐡)
69 simprr 771 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ 𝑦 ∈ 𝐡)
70 ovex 7386 . . . . . . . . . . . 12 (π‘₯ RngHomo 𝑦) ∈ V
7170a1i 11 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯ RngHomo 𝑦) ∈ V)
7271resiexd 7162 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ( I β†Ύ (π‘₯ RngHomo 𝑦)) ∈ V)
7364, 67, 68, 69, 72ovmpod 7503 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦) = ( I β†Ύ (π‘₯ RngHomo 𝑦)))
7473eqcomd 2742 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ( I β†Ύ (π‘₯ RngHomo 𝑦)) = (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦))
7563, 74coeq12d 5818 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦))) = (((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) ∘ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦)))
7634, 35, 75mpoeq123dva 7427 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∘ ( I β†Ύ (π‘₯ RngHomo 𝑦)))) = (π‘₯ ∈ (Baseβ€˜π‘…), 𝑦 ∈ (Baseβ€˜π‘…) ↦ (((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) ∘ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦))))
7733, 76eqtrd 2776 . . . . 5 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ (Baseβ€˜π‘…), 𝑦 ∈ (Baseβ€˜π‘…) ↦ (((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) ∘ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦))))
7817, 77opeq12d 4836 . . . 4 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ = ⟨((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)), (π‘₯ ∈ (Baseβ€˜π‘…), 𝑦 ∈ (Baseβ€˜π‘…) ↦ (((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) ∘ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦)))⟩)
79 eqid 2736 . . . . 5 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
80 eqid 2736 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
81 eqidd 2737 . . . . . 6 (πœ‘ β†’ ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡))
82 eqidd 2737 . . . . . 6 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))))
836, 80, 7, 8, 81, 82rngcifuestrc 46227 . . . . 5 (πœ‘ β†’ ( I β†Ύ 𝐡)(𝑅 Func (ExtStrCatβ€˜π‘ˆ))(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))))
84 funcrngcsetcALT.s . . . . . 6 𝑆 = (SetCatβ€˜π‘ˆ)
85 eqid 2736 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
86 eqid 2736 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
8780, 8estrcbas 18004 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)))
8887mpteq1d 5198 . . . . . 6 (πœ‘ β†’ (𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) = (𝑒 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ (Baseβ€˜π‘’)))
89 fveq2 6839 . . . . . . . . . . 11 (𝑀 = 𝑒 β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘’))
9089oveq2d 7369 . . . . . . . . . 10 (𝑀 = 𝑒 β†’ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)) = ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘’)))
9190reseq2d 5935 . . . . . . . . 9 (𝑀 = 𝑒 β†’ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))) = ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘’))))
92 fveq2 6839 . . . . . . . . . . 11 (𝑧 = 𝑣 β†’ (Baseβ€˜π‘§) = (Baseβ€˜π‘£))
9392oveq1d 7368 . . . . . . . . . 10 (𝑧 = 𝑣 β†’ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘’)) = ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’)))
9493reseq2d 5935 . . . . . . . . 9 (𝑧 = 𝑣 β†’ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘’))) = ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))))
9591, 94cbvmpov 7448 . . . . . . . 8 (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))) = (𝑒 ∈ π‘ˆ, 𝑣 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))))
9695a1i 11 . . . . . . 7 (πœ‘ β†’ (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))) = (𝑒 ∈ π‘ˆ, 𝑣 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’)))))
97 eqidd 2737 . . . . . . . 8 (πœ‘ β†’ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))) = ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))))
9887, 87, 97mpoeq123dv 7428 . . . . . . 7 (πœ‘ β†’ (𝑒 ∈ π‘ˆ, 𝑣 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’)))) = (𝑒 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)), 𝑣 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’)))))
9996, 98eqtrd 2776 . . . . . 6 (πœ‘ β†’ (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))) = (𝑒 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)), 𝑣 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’)))))
10080, 84, 85, 86, 8, 88, 99funcestrcsetc 18029 . . . . 5 (πœ‘ β†’ (𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’))((ExtStrCatβ€˜π‘ˆ) Func 𝑆)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))))
10179, 83, 100cofuval2 17765 . . . 4 (πœ‘ β†’ (⟨(𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)), (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))⟩ ∘func ⟨( I β†Ύ 𝐡), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))⟩) = ⟨((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)), (π‘₯ ∈ (Baseβ€˜π‘…), 𝑦 ∈ (Baseβ€˜π‘…) ↦ (((( I β†Ύ 𝐡)β€˜π‘₯)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))(( I β†Ύ 𝐡)β€˜π‘¦)) ∘ (π‘₯(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))𝑦)))⟩)
10278, 101eqtr4d 2779 . . 3 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ = (⟨(𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)), (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))⟩ ∘func ⟨( I β†Ύ 𝐡), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))⟩))
103 df-br 5104 . . . . 5 (( I β†Ύ 𝐡)(𝑅 Func (ExtStrCatβ€˜π‘ˆ))(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔))) ↔ ⟨( I β†Ύ 𝐡), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCatβ€˜π‘ˆ)))
10483, 103sylib 217 . . . 4 (πœ‘ β†’ ⟨( I β†Ύ 𝐡), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCatβ€˜π‘ˆ)))
105 df-br 5104 . . . . 5 ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’))((ExtStrCatβ€˜π‘ˆ) Func 𝑆)(𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€)))) ↔ ⟨(𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)), (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))⟩ ∈ ((ExtStrCatβ€˜π‘ˆ) Func 𝑆))
106100, 105sylib 217 . . . 4 (πœ‘ β†’ ⟨(𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)), (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))⟩ ∈ ((ExtStrCatβ€˜π‘ˆ) Func 𝑆))
107104, 106cofucl 17766 . . 3 (πœ‘ β†’ (⟨(𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)), (𝑀 ∈ π‘ˆ, 𝑧 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘§) ↑m (Baseβ€˜π‘€))))⟩ ∘func ⟨( I β†Ύ 𝐡), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ ( I β†Ύ (𝑓 RngHomo 𝑔)))⟩) ∈ (𝑅 Func 𝑆))
108102, 107eqeltrd 2838 . 2 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109 df-br 5104 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110108, 109sylibr 233 1 (πœ‘ β†’ 𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3443   ∩ cin 3907  βŸ¨cop 4590   class class class wbr 5103   ↦ cmpt 5186   I cid 5528   β†Ύ cres 5633   ∘ ccom 5635  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355   ↑m cmap 8761  WUnicwun 10632  Basecbs 17075   Func cfunc 17732   ∘func ccofu 17734  SetCatcsetc 17953  ExtStrCatcestrc 18001  Rngcrng 46104   RngHomo crngh 46115  RngCatcrngc 46187
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 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
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 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-pm 8764  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-wun 10634  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-fz 13417  df-struct 17011  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-ress 17105  df-plusg 17138  df-hom 17149  df-cco 17150  df-0g 17315  df-cat 17540  df-cid 17541  df-homf 17542  df-ssc 17685  df-resc 17686  df-subc 17687  df-func 17736  df-idfu 17737  df-cofu 17738  df-full 17783  df-fth 17784  df-setc 17954  df-estrc 18002  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-mhm 18593  df-grp 18743  df-ghm 18997  df-abl 19556  df-mgp 19888  df-mgmhm 46005  df-rng0 46105  df-rnghomo 46117  df-rngc 46189
This theorem is referenced by: (None)
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