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Theorem fuciso 17799
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 17783 . . . . 5 (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
3 fuciso.n . . . . . 6 𝑁 = (𝐢 Nat 𝐷)
41, 3fuchom 17784 . . . . 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 17684 . . . . . . . 8 (𝐹 ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 496 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
108simprd 497 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ Cat)
111, 9, 10fuccat 17794 . . . . 5 (πœ‘ β†’ 𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 17594 . . . 4 (πœ‘ β†’ (𝐹𝐼𝐺) βŠ† (𝐹𝑁𝐺))
1413sselda 3943 . . 3 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2738 . . . . 5 (Baseβ€˜π·) = (Baseβ€˜π·)
16 eqid 2738 . . . . 5 (Invβ€˜π·) = (Invβ€˜π·)
1710ad2antrr 725 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
19 relfunc 17683 . . . . . . . . 9 Rel (𝐢 Func 𝐷)
20 1st2ndbr 7964 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐹 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2119, 6, 20sylancr 588 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΉ)(𝐢 Func 𝐷)(2nd β€˜πΉ))
2218, 15, 21funcf1 17687 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2322adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΉ):𝐡⟢(Baseβ€˜π·))
2423ffvelcdmda 7030 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ π‘₯ ∈ 𝐡) β†’ ((1st β€˜πΉ)β€˜π‘₯) ∈ (Baseβ€˜π·))
25 1st2ndbr 7964 . . . . . . . . 9 ((Rel (𝐢 Func 𝐷) ∧ 𝐺 ∈ (𝐢 Func 𝐷)) β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2619, 12, 25sylancr 588 . . . . . . . 8 (πœ‘ β†’ (1st β€˜πΊ)(𝐢 Func 𝐷)(2nd β€˜πΊ))
2718, 15, 26funcf1 17687 . . . . . . 7 (πœ‘ β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2827adantr 482 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ (1st β€˜πΊ):𝐡⟢(Baseβ€˜π·))
2928ffvelcdmda 7030 . . . . 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 2738 . . . . . . . . . . . 12 (Invβ€˜π‘„) = (Invβ€˜π‘„)
322, 31, 11, 6, 12, 5isoval 17583 . . . . . . . . . . 11 (πœ‘ β†’ (𝐹𝐼𝐺) = dom (𝐹(Invβ€˜π‘„)𝐺))
3332eleq2d 2824 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Invβ€˜π‘„)𝐺)))
342, 31, 11, 6, 12invfun 17582 . . . . . . . . . . 11 (πœ‘ β†’ Fun (𝐹(Invβ€˜π‘„)𝐺))
35 funfvbrb 6997 . . . . . . . . . . 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 478 . . . . . . . 8 ((πœ‘ ∧ 𝐴 ∈ (𝐹𝐼𝐺)) β†’ 𝐴(𝐹(Invβ€˜π‘„)𝐺)((𝐹(Invβ€˜π‘„)𝐺)β€˜π΄))
391, 18, 3, 6, 12, 31, 16fucinv 17797 . . . . . . . . 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 482 . . . . . . . 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 1145 . . . . . 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 3233 . . . . 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 17584 . . . 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 513 . 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 482 . . 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 482 . . 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 482 . . 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 770 . . . 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 725 . . . . 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 482 . . . . . 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 7030 . . . . 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 482 . . . . . 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 7030 . . . . 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 772 . . . . . 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 6838 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π΄β€˜π‘₯) = (π΄β€˜π‘¦))
58 fveq2 6838 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΉ)β€˜π‘₯) = ((1st β€˜πΉ)β€˜π‘¦))
59 fveq2 6838 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((1st β€˜πΊ)β€˜π‘₯) = ((1st β€˜πΊ)β€˜π‘¦))
6058, 59oveq12d 7368 . . . . . . . 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 2833 . . . . . . 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 3579 . . . . . 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 581 . . . . 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 17609 . . . 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 17798 . . 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 17584 . 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 800 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3063   class class class wbr 5104   ↦ cmpt 5187  dom cdm 5631  Rel wrel 5636  Fun wfun 6486  βŸΆwf 6488  β€˜cfv 6492  (class class class)co 7350  1st c1st 7910  2nd c2nd 7911  Basecbs 17018  Catccat 17479  Invcinv 17563  Isociso 17564   Func cfunc 17675   Nat cnat 17763   FuncCat cfuc 17764
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 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  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 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-map 8701  df-ixp 8770  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-2 12150  df-3 12151  df-4 12152  df-5 12153  df-6 12154  df-7 12155  df-8 12156  df-9 12157  df-n0 12348  df-z 12434  df-dec 12552  df-uz 12697  df-fz 13354  df-struct 16954  df-slot 16989  df-ndx 17001  df-base 17019  df-hom 17092  df-cco 17093  df-cat 17483  df-cid 17484  df-sect 17565  df-inv 17566  df-iso 17567  df-func 17679  df-nat 17765  df-fuc 17766
This theorem is referenced by:  yonffthlem  18106
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