MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invfuc Structured version   Visualization version   GIF version

Theorem invfuc 17927
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 17813 . . . . . . . . . . . . 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 17812 . . . . . . . . . . . . 13 Rel (𝐢 Func 𝐷)
12 1st2ndbr 8028 . . . . . . . . . . . . 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 17816 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
1514ffvelcdmda 7087 . . . . . . . . . 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 8028 . . . . . . . . . . . . 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 17816 . . . . . . . . . . 11 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2019ffvelcdmda 7087 . . . . . . . . . 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 17708 . . . . . . . . 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 5192 . . . . . . . 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 5726 . . . . . . . 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 3147 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 𝑋 ∈ (((1st β€˜πΊ)β€˜π‘₯)(Hom β€˜π·)((1st β€˜πΉ)β€˜π‘₯)))
2910fvexi 6906 . . . . . 6 𝐡 ∈ V
30 mptelixpg 8929 . . . . . 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 6892 . . . . . 6 (π‘₯ = 𝑦 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘¦))
34 fveq2 6892 . . . . . 6 (π‘₯ = 𝑦 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘¦))
3533, 34oveq12d 7427 . . . . 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 8909 . . . 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 7010 . . . . . . . . . . . . . . . . . 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 5177 . . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . . . 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 6903 . . . . . . . . . . . . . . . 16 β„²π‘₯((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§)
4946, 47, 48nfbr 5196 . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ (π‘ˆβ€˜π‘₯) = (π‘ˆβ€˜π‘§))
51 fveq2 6892 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘§))
52 fveq2 6892 . . . . . . . . . . . . . . . . 17 (π‘₯ = 𝑧 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘§))
5351, 52oveq12d 7427 . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . 16 (π‘₯ = 𝑧 β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘§))
5550, 53, 54breq123d 5163 . . . . . . . . . . . . . . 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 3601 . . . . . . . . . . . . . 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 7088 . . . . . . . . . . . . . 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 7088 . . . . . . . . . . . . . 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 17707 . . . . . . . . . . . . 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 17700 . . . . . . . . . . 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 7424 . . . . . . . 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 7088 . . . . . . . . 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 17818 . . . . . . . . . 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 7088 . . . . . . . . 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 17627 . . . . . . . 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 17902 . . . . . . . . 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 17904 . . . . . . . 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 17630 . . . . . . 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 17906 . . . . . . . 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 7425 . . . . . . 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 7424 . . . . 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 7088 . . . . . 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 6903 . . . . . . . . . . . . 13 β„²π‘₯((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦)
9693, 94, 95nfbr 5196 . . . . . . . . . . . 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 6892 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ (π‘ˆβ€˜π‘₯) = (π‘ˆβ€˜π‘¦))
9834, 33oveq12d 7427 . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . 13 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘₯) = ((π‘₯ ∈ 𝐡 ↦ 𝑋)β€˜π‘¦))
10097, 98, 99breq123d 5163 . . . . . . . . . . . 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 3601 . . . . . . . . . . 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 17707 . . . . . . . . . 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 17700 . . . . . . . 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 17818 . . . . . . . 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 7088 . . . . . . 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 17629 . . . . . 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 17630 . . . . 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 17904 . . . . . . . 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 17630 . . . . . . 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 7425 . . . . . . 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 17628 . . . . . . 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 7425 . . . . 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 3204 . . 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 17899 . . 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 3304 . . 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 17926 . 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 3062  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   Γ— cxp 5675  Rel wrel 5682  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  Xcixp 8891  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609  Sectcsect 17691  Invcinv 17692   Func cfunc 17804   Nat cnat 17892   FuncCat cfuc 17893
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-cco 17222  df-cat 17612  df-cid 17613  df-sect 17694  df-inv 17695  df-func 17808  df-nat 17894  df-fuc 17895
This theorem is referenced by:  fuciso  17928  yonedainv  18234
  Copyright terms: Public domain W3C validator