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

Theorem fuciso 17938
Description: A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (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))
fuciso.i 𝐼 = (Isoβ€˜π‘„)
fuciso.j 𝐽 = (Isoβ€˜π·)
Assertion
Ref Expression
fuciso (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝐷   π‘₯,𝐼   π‘₯,𝐹   π‘₯,𝐺   π‘₯,𝐽   π‘₯,𝑁   πœ‘,π‘₯   π‘₯,𝑄

Proof of Theorem fuciso
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fuciso.q . . . . . 6 𝑄 = (𝐢 FuncCat 𝐷)
21fucbas 17922 . . . . 5 (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
3 fuciso.n . . . . . 6 𝑁 = (𝐢 Nat 𝐷)
41, 3fuchom 17923 . . . . 5 𝑁 = (Hom β€˜π‘„)
5 fuciso.i . . . . 5 𝐼 = (Isoβ€˜π‘„)
6 fuciso.f . . . . . . . 8 (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))
7 funcrcl 17820 . . . . . . . 8 (𝐹 ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 494 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
108simprd 495 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ Cat)
111, 9, 10fuccat 17933 . . . . 5 (πœ‘ β†’ 𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 17730 . . . 4 (πœ‘ β†’ (𝐹𝐼𝐺) βŠ† (𝐹𝑁𝐺))
1413sselda 3977 . . 3 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2726 . . . . 5 (Baseβ€˜π·) = (Baseβ€˜π·)
16 eqid 2726 . . . . 5 (Invβ€˜π·) = (Invβ€˜π·)
1710ad2antrr 723 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
19 relfunc 17819 . . . . . . . . 9 Rel (𝐢 Func 𝐷)
20 1st2ndbr 8024 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐹 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2119, 6, 20sylancr 586 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2218, 15, 21funcf1 17823 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2322adantr 480 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2423ffvelcdmda 7079 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΉ)β€˜π‘₯) ∈ (Baseβ€˜π·))
25 1st2ndbr 8024 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐺 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2619, 12, 25sylancr 586 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2718, 15, 26funcf1 17823 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2827adantr 480 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2928ffvelcdmda 7079 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΊ)β€˜π‘₯) ∈ (Baseβ€˜π·))
30 fuciso.j . . . . 5 𝐽 = (Isoβ€˜π·)
31 eqid 2726 . . . . . . . . . . . 12 (Invβ€˜π‘„) = (Invβ€˜π‘„)
322, 31, 11, 6, 12, 5isoval 17719 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹𝐼𝐺) = dom (𝐹(Invβ€˜π‘„)𝐺))
3332eleq2d 2813 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Invβ€˜π‘„)𝐺)))
342, 31, 11, 6, 12invfun 17718 . . . . . . . . . . 11 (πœ‘ β†’ Fun (𝐹(Invβ€˜π‘„)𝐺))
35 funfvbrb 7045 . . . . . . . . . . 11 (Fun (𝐹(Invβ€˜π‘„)𝐺) β†’ (𝐴 ∈ dom (𝐹(Invβ€˜π‘„)𝐺) ↔ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)))
3634, 35syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ dom (𝐹(Invβ€˜π‘„)𝐺) ↔ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)))
3733, 36bitrd 279 . . . . . . . . 9 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)))
3837biimpa 476 . . . . . . . 8 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄))
391, 18, 3, 6, 12, 31, 16fucinv 17936 . . . . . . . . 9 (πœ‘ β†’ (𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄) ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘₯))(((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)β€˜π‘₯))))
4039adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄) ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘₯))(((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)β€˜π‘₯))))
4138, 40mpbid 231 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄) ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘₯))(((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)β€˜π‘₯)))
4241simp3d 1141 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘₯))(((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)β€˜π‘₯))
4342r19.21bi 3242 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ (π΄β€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘₯))(((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)β€˜π‘₯))
4415, 16, 17, 24, 29, 30, 43inviso1 17720 . . . 4 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))
4544ralrimiva 3140 . . 3 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))
4614, 45jca 511 . 2 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))))
4711adantr 480 . . 3 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝑄 ∈ Cat)
486adantr 480 . . 3 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝐹 ∈ (𝐢 Func 𝐷))
4912adantr 480 . . 3 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝐺 ∈ (𝐢 Func 𝐷))
50 simprl 768 . . . 4 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝐴 ∈ (𝐹𝑁𝐺))
5110ad2antrr 723 . . . . 5 (((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) ∧ 𝑦 ∈ 𝐡) β†’ 𝐷 ∈ Cat)
5222adantr 480 . . . . . 6 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
5352ffvelcdmda 7079 . . . . 5 (((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) ∧ 𝑦 ∈ 𝐡) β†’ ((1st β€˜πΉ)β€˜π‘¦) ∈ (Baseβ€˜π·))
5427adantr 480 . . . . . 6 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
5554ffvelcdmda 7079 . . . . 5 (((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) ∧ 𝑦 ∈ 𝐡) β†’ ((1st β€˜πΊ)β€˜π‘¦) ∈ (Baseβ€˜π·))
56 simprr 770 . . . . . 6 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))
57 fveq2 6884 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π΄β€˜π‘₯) = (π΄β€˜π‘¦))
58 fveq2 6884 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘¦))
59 fveq2 6884 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘¦))
6058, 59oveq12d 7422 . . . . . . . 8 (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‘¦)))
6157, 60eleq12d 2821 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) ↔ (π΄β€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦))))
6261rspccva 3605 . . . . . 6 ((βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) ∧ 𝑦 ∈ 𝐡) β†’ (π΄β€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦)))
6356, 62sylan 579 . . . . 5 (((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) ∧ 𝑦 ∈ 𝐡) β†’ (π΄β€˜π‘¦) ∈ (((1st β€˜πΉ)β€˜π‘¦)𝐽((1st β€˜πΊ)β€˜π‘¦)))
6415, 30, 16, 51, 53, 55, 63invisoinvr 17745 . . . 4 (((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) ∧ 𝑦 ∈ 𝐡) β†’ (π΄β€˜π‘¦)(((1st β€˜πΉ)β€˜π‘¦)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))((((1st β€˜πΉ)β€˜π‘¦)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))β€˜(π΄β€˜π‘¦)))
651, 18, 3, 48, 49, 31, 16, 50, 64invfuc 17937 . . 3 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝐴(𝐹(Invβ€˜π‘„)𝐺)(𝑦 ∈ 𝐡 ↦ ((((1st β€˜πΉ)β€˜π‘¦)(Invβ€˜π·)((1st β€˜πΊ)β€˜π‘¦))β€˜(π΄β€˜π‘¦))))
662, 31, 47, 48, 49, 5, 65inviso1 17720 . 2 ((πœ‘ ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))) β†’ 𝐴 ∈ (𝐹𝐼𝐺))
6746, 66impbida 798 1 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669  Rel wrel 5674  Fun wfun 6530  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  Catccat 17615  Invcinv 17699  Isociso 17700   Func cfunc 17811   Nat cnat 17902   FuncCat cfuc 17903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17087  df-slot 17122  df-ndx 17134  df-base 17152  df-hom 17228  df-cco 17229  df-cat 17619  df-cid 17620  df-sect 17701  df-inv 17702  df-iso 17703  df-func 17815  df-nat 17904  df-fuc 17905
This theorem is referenced by:  yonffthlem  18245
  Copyright terms: Public domain W3C validator