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Theorem funcres2c 17183
 Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
funcres2c.a 𝐴 = (Base‘𝐶)
funcres2c.e 𝐸 = (𝐷s 𝑆)
funcres2c.d (𝜑𝐷 ∈ Cat)
funcres2c.r (𝜑𝑆𝑉)
funcres2c.1 (𝜑𝐹:𝐴𝑆)
Assertion
Ref Expression
funcres2c (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))

Proof of Theorem funcres2c
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 864 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
21a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)))
3 olc 865 . . 3 (𝐹(𝐶 Func 𝐸)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
43a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func 𝐸)𝐺 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)))
5 funcres2c.a . . . . 5 𝐴 = (Base‘𝐶)
6 eqid 2798 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2798 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
8 eqid 2798 . . . . . . 7 (Homf𝐷) = (Homf𝐷)
9 funcres2c.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
10 inss2 4159 . . . . . . . 8 (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
1110a1i 11 . . . . . . 7 (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
127, 8, 9, 11fullsubc 17132 . . . . . 6 (𝜑 → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
1312adantr 484 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
148, 7homffn 16975 . . . . . . 7 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15 xpss12 5538 . . . . . . . 8 (((𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷) ∧ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)) → ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
1610, 10, 15mp2an 691 . . . . . . 7 ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17 fnssres 6450 . . . . . . 7 (((Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) ∧ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))) → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))
1814, 16, 17mp2an 691 . . . . . 6 ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))
1918a1i 11 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) Fn ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))
20 funcres2c.1 . . . . . . . 8 (𝜑𝐹:𝐴𝑆)
2120adantr 484 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴𝑆)
2221ffnd 6496 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
2321frnd 6502 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹𝑆)
24 simpr 488 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
255, 7, 24funcf1 17148 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
2625frnd 6502 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27 eqid 2798 . . . . . . . . . . 11 (Base‘𝐸) = (Base‘𝐸)
28 simpr 488 . . . . . . . . . . 11 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
295, 27, 28funcf1 17148 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
3029frnd 6502 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31 funcres2c.e . . . . . . . . . 10 𝐸 = (𝐷s 𝑆)
3231, 7ressbasss 16568 . . . . . . . . 9 (Base‘𝐸) ⊆ (Base‘𝐷)
3330, 32sstrdi 3929 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
3426, 33jaodan 955 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
3523, 34ssind 4162 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36 df-f 6336 . . . . . 6 (𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷))))
3722, 35, 36sylanbrc 586 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
38 eqid 2798 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
39 simpr 488 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
40 simplrl 776 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥𝐴)
41 simplrr 777 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦𝐴)
425, 6, 38, 39, 40, 41funcf2 17150 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
43 eqid 2798 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
44 simpr 488 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
45 simplrl 776 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥𝐴)
46 simplrr 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦𝐴)
475, 6, 43, 44, 45, 46funcf2 17150 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
48 funcres2c.r . . . . . . . . . . . . 13 (𝜑𝑆𝑉)
4931, 38resshom 16703 . . . . . . . . . . . . 13 (𝑆𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
5048, 49syl 17 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
5150ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
5251oveqd 7162 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
5352feq3d 6482 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦))))
5447, 53mpbird 260 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5542, 54jaodan 955 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5655an32s 651 . . . . . 6 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5737adantr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
58 simprl 770 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5957, 58ffvelrnd 6839 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60 simprr 772 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
6157, 60ffvelrnd 6839 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (𝑆 ∩ (Base‘𝐷)))
6259, 61ovresd 7306 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹𝑦)) = ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)))
6359elin2d 4129 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (Base‘𝐷))
6461elin2d 4129 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (Base‘𝐷))
658, 7, 38, 63, 64homfval 16974 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6662, 65eqtrd 2833 . . . . . . 7 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6766feq3d 6482 . . . . . 6 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
6856, 67mpbird 260 . . . . 5 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹𝑦)))
695, 6, 13, 19, 37, 68funcres2b 17179 . . . 4 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))𝐺))
70 eqidd 2799 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐶) = (Homf𝐶))
71 eqidd 2799 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐶) = (compf𝐶))
727ressinbas 16572 . . . . . . . . . . 11 (𝑆𝑉 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7348, 72syl 17 . . . . . . . . . 10 (𝜑 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7431, 73syl5eq 2845 . . . . . . . . 9 (𝜑𝐸 = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7574fveq2d 6659 . . . . . . . 8 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
76 eqid 2798 . . . . . . . . . 10 (𝐷s (𝑆 ∩ (Base‘𝐷))) = (𝐷s (𝑆 ∩ (Base‘𝐷)))
77 eqid 2798 . . . . . . . . . 10 (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) = (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))
787, 8, 9, 11, 76, 77fullresc 17133 . . . . . . . . 9 (𝜑 → ((Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))) ∧ (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))))
7978simpld 498 . . . . . . . 8 (𝜑 → (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8075, 79eqtrd 2833 . . . . . . 7 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8180adantr 484 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8274fveq2d 6659 . . . . . . . 8 (𝜑 → (compf𝐸) = (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
8378simprd 499 . . . . . . . 8 (𝜑 → (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8482, 83eqtrd 2833 . . . . . . 7 (𝜑 → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8584adantr 484 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
86 df-br 5035 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87 funcrcl 17145 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8886, 87sylbi 220 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8988simpld 498 . . . . . . . . 9 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
90 df-br 5035 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91 funcrcl 17145 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9290, 91sylbi 220 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9392simpld 498 . . . . . . . . 9 (𝐹(𝐶 Func 𝐸)𝐺𝐶 ∈ Cat)
9489, 93jaoi 854 . . . . . . . 8 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
9594elexd 3462 . . . . . . 7 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
9695adantl 485 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
9731ovexi 7179 . . . . . . 7 𝐸 ∈ V
9897a1i 11 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99 ovexd 7180 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) ∈ V)
10070, 71, 81, 85, 96, 96, 98, 99funcpropd 17182 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐶 Func 𝐸) = (𝐶 Func (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
101100breqd 5045 . . . 4 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐸)𝐺𝐹(𝐶 Func (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))))𝐺))
10269, 101bitr4d 285 . . 3 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
103102ex 416 . 2 (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)))
1042, 4, 103pm5.21ndd 384 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4534   class class class wbr 5034   × cxp 5521  ran crn 5524   ↾ cres 5525   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  Basecbs 16495   ↾s cress 16496  Hom chom 16588  Catccat 16947  Homf chomf 16949  compfccomf 16950   ↾cat cresc 17090  Subcatcsubc 17091   Func cfunc 17136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-map 8409  df-pm 8410  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-hom 16601  df-cco 16602  df-cat 16951  df-cid 16952  df-homf 16953  df-comf 16954  df-ssc 17092  df-resc 17093  df-subc 17094  df-func 17140 This theorem is referenced by:  fthres2c  17213  fullres2c  17221
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