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Theorem fuciso 17972
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 17956 . . . . 5 (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
3 fuciso.n . . . . . 6 𝑁 = (𝐢 Nat 𝐷)
41, 3fuchom 17957 . . . . 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 17854 . . . . . . . 8 (𝐹 ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 493 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
108simprd 494 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ Cat)
111, 9, 10fuccat 17967 . . . . 5 (πœ‘ β†’ 𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 17764 . . . 4 (πœ‘ β†’ (𝐹𝐼𝐺) βŠ† (𝐹𝑁𝐺))
1413sselda 3980 . . 3 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2727 . . . . 5 (Baseβ€˜π·) = (Baseβ€˜π·)
16 eqid 2727 . . . . 5 (Invβ€˜π·) = (Invβ€˜π·)
1710ad2antrr 724 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
19 relfunc 17853 . . . . . . . . 9 Rel (𝐢 Func 𝐷)
20 1st2ndbr 8050 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐹 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2119, 6, 20sylancr 585 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2218, 15, 21funcf1 17857 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2322adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2423ffvelcdmda 7097 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΉ)β€˜π‘₯) ∈ (Baseβ€˜π·))
25 1st2ndbr 8050 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐺 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2619, 12, 25sylancr 585 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2718, 15, 26funcf1 17857 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2827adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2928ffvelcdmda 7097 . . . . 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 2727 . . . . . . . . . . . 12 (Invβ€˜π‘„) = (Invβ€˜π‘„)
322, 31, 11, 6, 12, 5isoval 17753 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹𝐼𝐺) = dom (𝐹(Invβ€˜π‘„)𝐺))
3332eleq2d 2814 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Invβ€˜π‘„)𝐺)))
342, 31, 11, 6, 12invfun 17752 . . . . . . . . . . 11 (πœ‘ β†’ Fun (𝐹(Invβ€˜π‘„)𝐺))
35 funfvbrb 7063 . . . . . . . . . . 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 278 . . . . . . . . 9 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄)))
3837biimpa 475 . . . . . . . 8 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄))
391, 18, 3, 6, 12, 31, 16fucinv 17970 . . . . . . . . 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 479 . . . . . . . 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 3244 . . . . 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 17754 . . . 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 3142 . . 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 510 . 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 479 . . 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 479 . . 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 479 . . 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 769 . . . 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 724 . . . . 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 479 . . . . . 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 7097 . . . . 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 479 . . . . . 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 7097 . . . . 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 771 . . . . . 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 6900 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π΄β€˜π‘₯) = (π΄β€˜π‘¦))
58 fveq2 6900 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘¦))
59 fveq2 6900 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘¦))
6058, 59oveq12d 7442 . . . . . . . 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 2822 . . . . . . 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 3608 . . . . . 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 578 . . . . 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 17779 . . . 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 17971 . . 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 17754 . 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 799 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057   class class class wbr 5150   ↦ cmpt 5233  dom cdm 5680  Rel wrel 5685  Fun wfun 6545  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424  1st c1st 7995  2nd c2nd 7996  Basecbs 17185  Catccat 17649  Invcinv 17733  Isociso 17734   Func cfunc 17845   Nat cnat 17936   FuncCat cfuc 17937
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-map 8851  df-ixp 8921  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12509  df-z 12595  df-dec 12714  df-uz 12859  df-fz 13523  df-struct 17121  df-slot 17156  df-ndx 17168  df-base 17186  df-hom 17262  df-cco 17263  df-cat 17653  df-cid 17654  df-sect 17735  df-inv 17736  df-iso 17737  df-func 17849  df-nat 17938  df-fuc 17939
This theorem is referenced by:  yonffthlem  18279
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