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Theorem funcrngcsetcALT 46198
Description: Alternate proof of funcrngcsetc 46197, using cofuval2 17733 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 46196, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 17997. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 46197. (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 2793 . . . . . 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 46164 . . . . . . . . . . 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 2790 . . . . . 6 (πœ‘ β†’ (𝑒 ∈ 𝐡 ↦ (Baseβ€˜π‘’)) = ((𝑒 ∈ π‘ˆ ↦ (Baseβ€˜π‘’)) ∘ ( I β†Ύ 𝐡)))
174, 16eqtrd 2777 . . . . 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 2737 . . . . . . . . . . . . 13 (Baseβ€˜π‘₯) = (Baseβ€˜π‘₯)
21 eqid 2737 . . . . . . . . . . . . 13 (Baseβ€˜π‘¦) = (Baseβ€˜π‘¦)
2220, 21rnghmf 46098 . . . . . . . . . . . 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 8736 . . . . . . . . . . . . 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 2790 . . . . . . . 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 7428 . . . . . . 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 2777 . . . . . 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 7369 . . . . . . . . 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 2738 . . . . . . . . . 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 7369 . . . . . . . . . . 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 7384 . . . . . . . . . . . 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 7501 . . . . . . . . 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 2778 . . . . . . . 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 2738 . . . . . . . . . 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 7360 . . . . . . . . . . . 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 7384 . . . . . . . . . . . 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 7501 . . . . . . . . 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 2743 . . . . . . . 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 7425 . . . . . 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 2777 . . . . 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 2737 . . . . 5 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
80 eqid 2737 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
81 eqidd 2738 . . . . . 6 (πœ‘ β†’ ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡))
82 eqidd 2738 . . . . . 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 46196 . . . . 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 2737 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
86 eqid 2737 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
8780, 8estrcbas 17972 . . . . . . 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 7367 . . . . . . . . . 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 7366 . . . . . . . . . 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 7446 . . . . . . . 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 2738 . . . . . . . 8 (πœ‘ β†’ ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))) = ( I β†Ύ ((Baseβ€˜π‘£) ↑m (Baseβ€˜π‘’))))
9887, 87, 97mpoeq123dv 7426 . . . . . . 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 2777 . . . . . 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 17997 . . . . 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 17733 . . . 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 2780 . . 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 17734 . . 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 7351   ∈ cmpo 7353   ↑m cmap 8723  WUnicwun 10594  Basecbs 17043   Func cfunc 17700   ∘func ccofu 17702  SetCatcsetc 17921  ExtStrCatcestrc 17969  Rngcrng 46073   RngHomo crngh 46084  RngCatcrngc 46156
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  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 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-map 8725  df-pm 8726  df-ixp 8794  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-wun 10596  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-fz 13379  df-struct 16979  df-sets 16996  df-slot 17014  df-ndx 17026  df-base 17044  df-ress 17073  df-plusg 17106  df-hom 17117  df-cco 17118  df-0g 17283  df-cat 17508  df-cid 17509  df-homf 17510  df-ssc 17653  df-resc 17654  df-subc 17655  df-func 17704  df-idfu 17705  df-cofu 17706  df-full 17751  df-fth 17752  df-setc 17922  df-estrc 17970  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-mhm 18561  df-grp 18711  df-ghm 18965  df-abl 19524  df-mgp 19856  df-mgmhm 45974  df-rng0 46074  df-rnghomo 46086  df-rngc 46158
This theorem is referenced by: (None)
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