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Theorem invfuc 17868
Description: If 𝑉(π‘₯) is an inverse to π‘ˆ(π‘₯) for each π‘₯, and π‘ˆ is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q 𝑄 = (𝐢 FuncCat 𝐷)
fuciso.b 𝐡 = (Baseβ€˜πΆ)
fuciso.n 𝑁 = (𝐢 Nat 𝐷)
fuciso.f (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))
fuciso.g (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))
fucinv.i 𝐼 = (Invβ€˜π‘„)
fucinv.j 𝐽 = (Invβ€˜π·)
invfuc.u (πœ‘ β†’ π‘ˆ ∈ (𝐹𝑁𝐺))
invfuc.v ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))𝑋)
Assertion
Ref Expression
invfuc (πœ‘ β†’ π‘ˆ(𝐹𝐼𝐺)(π‘₯ ∈ 𝐡 ↦ 𝑋))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝐷   π‘₯,𝐼   π‘₯,𝐹   π‘₯,𝐺   π‘₯,𝐽   π‘₯,𝑁   πœ‘,π‘₯   π‘₯,𝑄   π‘₯,π‘ˆ
Allowed substitution hint:   𝑋(π‘₯)

Proof of Theorem invfuc
Dummy variables 𝑦 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfuc.u . 2 (πœ‘ β†’ π‘ˆ ∈ (𝐹𝑁𝐺))
2 invfuc.v . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))𝑋)
3 eqid 2733 . . . . . . . . . 10 (Baseβ€˜π·) = (Baseβ€˜π·)
4 fucinv.j . . . . . . . . . 10 𝐽 = (Invβ€˜π·)
5 fuciso.f . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))
6 funcrcl 17754 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
75, 6syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
87simprd 497 . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ Cat)
98adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ 𝐷 ∈ Cat)
10 fuciso.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΆ)
11 relfunc 17753 . . . . . . . . . . . . 13 Rel (𝐢 Func 𝐷)
12 1st2ndbr 7975 . . . . . . . . . . . . 13 ((Rel (𝐢 Func 𝐷) ∧ 𝐹 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
1311, 5, 12sylancr 588 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
1410, 3, 13funcf1 17757 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
1514ffvelcdmda 7036 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΉ)β€˜π‘₯) ∈ (Baseβ€˜π·))
16 fuciso.g . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))
17 1st2ndbr 7975 . . . . . . . . . . . . 13 ((Rel (𝐢 Func 𝐷) ∧ 𝐺 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
1811, 16, 17sylancr 588 . . . . . . . . . . . 12 (πœ‘ β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
1910, 3, 18funcf1 17757 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2019ffvelcdmda 7036 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΊ)β€˜π‘₯) ∈ (Baseβ€˜π·))
21 eqid 2733 . . . . . . . . . 10 (Hom β€˜π·) = (Hom β€˜π·)
223, 4, 9, 15, 20, 21invss 17649 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) βŠ† ((((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) Γ— (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯))))
2322ssbrd 5149 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))𝑋 β†’ (π‘ˆβ€˜π‘₯)((((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) Γ— (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))𝑋))
242, 23mpd 15 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘ˆβ€˜π‘₯)((((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) Γ— (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))𝑋)
25 brxp 5682 . . . . . . . 8 ((π‘ˆβ€˜π‘₯)((((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) Γ— (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))𝑋 ↔ ((π‘ˆβ€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) ∧ 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯))))
2625simprbi 498 . . . . . . 7 ((π‘ˆβ€˜π‘₯)((((1st β€˜πΉ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘₯)) Γ— (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))𝑋 β†’ 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
2724, 26syl 17 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
2827ralrimiva 3140 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
2910fvexi 6857 . . . . . 6 𝐡 ∈ V
30 mptelixpg 8876 . . . . . 6 (𝐡 ∈ V β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ Xπ‘₯ ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)) ↔ βˆ€π‘₯ ∈ 𝐡 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯))))
3129, 30ax-mp 5 . . . . 5 ((π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ Xπ‘₯ ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)) ↔ βˆ€π‘₯ ∈ 𝐡 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
3228, 31sylibr 233 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ Xπ‘₯ ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
33 fveq2 6843 . . . . . 6 (π‘₯ = 𝑦 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘¦))
34 fveq2 6843 . . . . . 6 (π‘₯ = 𝑦 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘¦))
3533, 34oveq12d 7376 . . . . 5 (π‘₯ = 𝑦 β†’ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)) = (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)))
3635cbvixpv 8856 . . . 4 Xπ‘₯ ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)) = X𝑦 ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦))
3732, 36eleqtrdi 2844 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ X𝑦 ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)))
38 simpr2 1196 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ 𝑧 ∈ 𝐡)
39 simpr 486 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
40 eqid 2733 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ 𝐡 ↦ 𝑋) = (π‘₯ ∈ 𝐡 ↦ 𝑋)
4140fvmpt2 6960 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ 𝐡 ∧ 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯))) β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = 𝑋)
4239, 27, 41syl2anc 585 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = 𝑋)
432, 42breqtrrd 5134 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯))
4443ralrimiva 3140 . . . . . . . . . . . . . . 15 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯))
4544adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯))
46 nfcv 2904 . . . . . . . . . . . . . . . 16 β„²π‘₯(π‘ˆβ€˜π‘§)
47 nfcv 2904 . . . . . . . . . . . . . . . 16 β„²π‘₯(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))
48 nffvmpt1 6854 . . . . . . . . . . . . . . . 16 β„²π‘₯((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)
4946, 47, 48nfbr 5153 . . . . . . . . . . . . . . 15 β„²π‘₯(π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)
50 fveq2 6843 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ (π‘ˆβ€˜π‘₯) = (π‘ˆβ€˜π‘§))
51 fveq2 6843 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘§))
52 fveq2 6843 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘§))
5351, 52oveq12d 7376 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) = (((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§)))
54 fveq2 6843 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§))
5550, 53, 54breq123d 5120 . . . . . . . . . . . . . . 15 (π‘₯ = 𝑧 β†’ ((π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) ↔ (π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)))
5649, 55rspc 3568 . . . . . . . . . . . . . 14 (𝑧 ∈ 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) β†’ (π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)))
5738, 45, 56sylc 65 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§))
588adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ 𝐷 ∈ Cat)
5914adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
6059, 38ffvelcdmd 7037 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((1st β€˜πΉ)β€˜π‘§) ∈ (Baseβ€˜π·))
6119adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
6261, 38ffvelcdmd 7037 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((1st β€˜πΊ)β€˜π‘§) ∈ (Baseβ€˜π·))
63 eqid 2733 . . . . . . . . . . . . . 14 (Sectβ€˜π·) = (Sectβ€˜π·)
643, 4, 58, 60, 62, 63isinv 17648 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)𝐽((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ↔ ((π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(((1st β€˜πΊ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§))))
6557, 64mpbid 231 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(((1st β€˜πΊ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§)))
6665simpld 496 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§))
67 eqid 2733 . . . . . . . . . . . 12 (compβ€˜π·) = (compβ€˜π·)
68 eqid 2733 . . . . . . . . . . . 12 (Idβ€˜π·) = (Idβ€˜π·)
693, 21, 67, 68, 63, 58, 60, 62issect 17641 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘§)(((1st β€˜πΉ)β€˜π‘§)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ↔ ((π‘ˆβ€˜π‘§) ∈ (((1st β€˜πΉ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ∈ (((1st β€˜πΊ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘§)) ∧ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§)) = ((Idβ€˜π·)β€˜((1st β€˜πΉ)β€˜π‘§)))))
7066, 69mpbid 231 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘§) ∈ (((1st β€˜πΉ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ∈ (((1st β€˜πΊ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘§)) ∧ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§)) = ((Idβ€˜π·)β€˜((1st β€˜πΉ)β€˜π‘§))))
7170simp3d 1145 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§)) = ((Idβ€˜π·)β€˜((1st β€˜πΉ)β€˜π‘§)))
7271oveq1d 7373 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§))(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)) = (((Idβ€˜π·)β€˜((1st β€˜πΉ)β€˜π‘§))(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)))
73 simpr1 1195 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ 𝑦 ∈ 𝐡)
7459, 73ffvelcdmd 7037 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((1st β€˜πΉ)β€˜π‘¦) ∈ (Baseβ€˜π·))
75 eqid 2733 . . . . . . . . . . 11 (Hom β€˜πΆ) = (Hom β€˜πΆ)
7613adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
7710, 75, 21, 76, 73, 38funcf2 17759 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (𝑦(2nd β€˜πΉ)𝑧):(𝑦(Hom β€˜πΆ)𝑧)⟢(((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘§)))
78 simpr3 1197 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))
7977, 78ffvelcdmd 7037 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘§)))
803, 21, 68, 58, 74, 67, 60, 79catlid 17568 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((Idβ€˜π·)β€˜((1st β€˜πΉ)β€˜π‘§))(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)) = ((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“))
8172, 80eqtr2d 2774 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“) = ((((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§))(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)))
82 fuciso.n . . . . . . . . 9 𝑁 = (𝐢 Nat 𝐷)
831adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ π‘ˆ ∈ (𝐹𝑁𝐺))
8482, 83nat1st2nd 17843 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ π‘ˆ ∈ (⟨(1st β€˜πΉ), (2nd β€˜πΉ)βŸ©π‘βŸ¨(1st β€˜πΊ), (2nd β€˜πΊ)⟩))
8582, 84, 10, 21, 38natcl 17845 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘§) ∈ (((1st β€˜πΉ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)))
8670simp2d 1144 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§) ∈ (((1st β€˜πΊ)β€˜π‘§)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘§)))
873, 21, 67, 58, 74, 60, 62, 79, 85, 60, 86catass 17571 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘§), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(π‘ˆβ€˜π‘§))(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)) = (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘ˆβ€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“))))
8882, 84, 10, 75, 67, 73, 38, 78nati 17847 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)) = (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦)))
8988oveq2d 7374 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘ˆβ€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“))) = (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))))
9081, 87, 893eqtrd 2777 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“) = (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))))
9190oveq1d 7373 . . . . 5 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = ((((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦)))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))
9261, 73ffvelcdmd 7037 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((1st β€˜πΊ)β€˜π‘¦) ∈ (Baseβ€˜π·))
93 nfcv 2904 . . . . . . . . . . . . 13 β„²π‘₯(π‘ˆβ€˜π‘¦)
94 nfcv 2904 . . . . . . . . . . . . 13 β„²π‘₯(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))
95 nffvmpt1 6854 . . . . . . . . . . . . 13 β„²π‘₯((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)
9693, 94, 95nfbr 5153 . . . . . . . . . . . 12 β„²π‘₯(π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)
97 fveq2 6843 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (π‘ˆβ€˜π‘₯) = (π‘ˆβ€˜π‘¦))
9834, 33oveq12d 7376 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) = (((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦)))
99 fveq2 6843 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))
10097, 98, 99breq123d 5120 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) ↔ (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))
10196, 100rspc 3568 . . . . . . . . . . 11 (𝑦 ∈ 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) β†’ (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))
10273, 45, 101sylc 65 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))
1033, 4, 58, 74, 92, 63isinv 17648 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ↔ ((π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)(((1st β€˜πΊ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘¦))(π‘ˆβ€˜π‘¦))))
104102, 103mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ∧ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)(((1st β€˜πΊ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘¦))(π‘ˆβ€˜π‘¦)))
105104simprd 497 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)(((1st β€˜πΊ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘¦))(π‘ˆβ€˜π‘¦))
1063, 21, 67, 68, 63, 58, 92, 74issect 17641 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)(((1st β€˜πΊ)β€˜π‘¦)(Sectβ€˜π·)((1st β€˜πΉ)β€˜π‘¦))(π‘ˆβ€˜π‘¦) ↔ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ∈ (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)) ∧ (π‘ˆβ€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘¦)) ∧ ((π‘ˆβ€˜π‘¦)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = ((Idβ€˜π·)β€˜((1st β€˜πΊ)β€˜π‘¦)))))
107105, 106mpbid 231 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ∈ (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)) ∧ (π‘ˆβ€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘¦)) ∧ ((π‘ˆβ€˜π‘¦)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = ((Idβ€˜π·)β€˜((1st β€˜πΊ)β€˜π‘¦))))
108107simp1d 1143 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦) ∈ (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)))
109107simp2d 1144 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘¦)))
11018adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
11110, 75, 21, 110, 73, 38funcf2 17759 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (𝑦(2nd β€˜πΊ)𝑧):(𝑦(Hom β€˜πΆ)𝑧)⟢(((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)))
112111, 78ffvelcdmd 7037 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“) ∈ (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)))
1133, 21, 67, 58, 74, 92, 62, 109, 112catcocl 17570 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦)) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘§)))
1143, 21, 67, 58, 92, 74, 62, 108, 113, 60, 86catass 17571 . . . . 5 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))(((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦)))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))))
11582, 84, 10, 21, 73natcl 17845 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (π‘ˆβ€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΊ)β€˜π‘¦)))
1163, 21, 67, 58, 92, 74, 92, 108, 115, 62, 112catass 17571 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘ˆβ€˜π‘¦)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))))
117107simp3d 1145 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((π‘ˆβ€˜π‘¦)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = ((Idβ€˜π·)β€˜((1st β€˜πΊ)β€˜π‘¦)))
118117oveq2d 7374 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘ˆβ€˜π‘¦)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))) = (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((Idβ€˜π·)β€˜((1st β€˜πΊ)β€˜π‘¦))))
1193, 21, 68, 58, 92, 67, 62, 112catrid 17569 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((Idβ€˜π·)β€˜((1st β€˜πΊ)β€˜π‘¦))) = ((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“))
120116, 118, 1193eqtrd 2777 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ ((((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)) = ((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“))
121120oveq2d 7374 . . . . 5 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)(⟨((1st β€˜πΉ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))(π‘ˆβ€˜π‘¦))(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΊ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))) = (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)))
12291, 114, 1213eqtrrd 2778 . . . 4 ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑦(Hom β€˜πΆ)𝑧))) β†’ (((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)) = (((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))
123122ralrimivvva 3197 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 βˆ€π‘“ ∈ (𝑦(Hom β€˜πΆ)𝑧)(((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)) = (((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))
12482, 10, 75, 21, 67, 16, 5isnat2 17840 . . 3 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ (𝐺𝑁𝐹) ↔ ((π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ X𝑦 ∈ 𝐡 (((1st β€˜πΊ)β€˜π‘¦)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘¦)) ∧ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 βˆ€π‘“ ∈ (𝑦(Hom β€˜πΆ)𝑧)(((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΊ)β€˜π‘§)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((𝑦(2nd β€˜πΊ)𝑧)β€˜π‘“)) = (((𝑦(2nd β€˜πΉ)𝑧)β€˜π‘“)(⟨((1st β€˜πΊ)β€˜π‘¦), ((1st β€˜πΉ)β€˜π‘¦)⟩(compβ€˜π·)((1st β€˜πΉ)β€˜π‘§))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)))))
12537, 123, 124mpbir2and 712 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ (𝐺𝑁𝐹))
126 nfv 1918 . . . 4 Ⅎ𝑦(π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯)
127126, 96, 100cbvralw 3288 . . 3 (βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) ↔ βˆ€π‘¦ ∈ 𝐡 (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))
12844, 127sylib 217 . 2 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))
129 fuciso.q . . 3 𝑄 = (𝐢 FuncCat 𝐷)
130 fucinv.i . . 3 𝐼 = (Invβ€˜π‘„)
131129, 10, 82, 5, 16, 130, 4fucinv 17867 . 2 (πœ‘ β†’ (π‘ˆ(𝐹𝐼𝐺)(π‘₯ ∈ 𝐡 ↦ 𝑋) ↔ (π‘ˆ ∈ (𝐹𝑁𝐺) ∧ (π‘₯ ∈ 𝐡 ↦ 𝑋) ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘¦ ∈ 𝐡 (π‘ˆβ€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))))
1321, 125, 128, 131mpbir3and 1343 1 (πœ‘ β†’ π‘ˆ(𝐹𝐼𝐺)(π‘₯ ∈ 𝐡 ↦ 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444  βŸ¨cop 4593   class class class wbr 5106   ↦ cmpt 5189   Γ— cxp 5632  Rel wrel 5639  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  Xcixp 8838  Basecbs 17088  Hom chom 17149  compcco 17150  Catccat 17549  Idccid 17550  Sectcsect 17632  Invcinv 17633   Func cfunc 17745   Nat cnat 17833   FuncCat cfuc 17834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-sect 17635  df-inv 17636  df-func 17749  df-nat 17835  df-fuc 17836
This theorem is referenced by:  fuciso  17869  yonedainv  18175
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