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Theorem funcres2c 17954
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 867 . . 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 868 . . 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 2734 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2734 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
8 eqid 2734 . . . . . . 7 (Homf𝐷) = (Homf𝐷)
9 funcres2c.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
10 inss2 4245 . . . . . . . 8 (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
1110a1i 11 . . . . . . 7 (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
127, 8, 9, 11fullsubc 17900 . . . . . 6 (𝜑 → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
1312adantr 480 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
148, 7homffn 17737 . . . . . . 7 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15 xpss12 5703 . . . . . . . 8 (((𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷) ∧ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)) → ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
1610, 10, 15mp2an 692 . . . . . . 7 ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17 fnssres 6691 . . . . . . 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 692 . . . . . 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 480 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴𝑆)
2221ffnd 6737 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
2321frnd 6744 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹𝑆)
24 simpr 484 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
255, 7, 24funcf1 17916 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
2625frnd 6744 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27 eqid 2734 . . . . . . . . . . 11 (Base‘𝐸) = (Base‘𝐸)
28 simpr 484 . . . . . . . . . . 11 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
295, 27, 28funcf1 17916 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
3029frnd 6744 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31 funcres2c.e . . . . . . . . . 10 𝐸 = (𝐷s 𝑆)
3231, 7ressbasss 17283 . . . . . . . . 9 (Base‘𝐸) ⊆ (Base‘𝐷)
3330, 32sstrdi 4007 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
3426, 33jaodan 959 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
3523, 34ssind 4248 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36 df-f 6566 . . . . . 6 (𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷))))
3722, 35, 36sylanbrc 583 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
38 eqid 2734 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
39 simpr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
40 simplrl 777 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥𝐴)
41 simplrr 778 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦𝐴)
425, 6, 38, 39, 40, 41funcf2 17918 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
43 eqid 2734 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
44 simpr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
45 simplrl 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥𝐴)
46 simplrr 778 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦𝐴)
475, 6, 43, 44, 45, 46funcf2 17918 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
48 funcres2c.r . . . . . . . . . . . . 13 (𝜑𝑆𝑉)
4931, 38resshom 17464 . . . . . . . . . . . . 13 (𝑆𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
5048, 49syl 17 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
5150ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
5251oveqd 7447 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
5352feq3d 6723 . . . . . . . . 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 257 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5542, 54jaodan 959 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5655an32s 652 . . . . . 6 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5737adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
58 simprl 771 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
5957, 58ffvelcdmd 7104 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
60 simprr 773 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
6157, 60ffvelcdmd 7104 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (𝑆 ∩ (Base‘𝐷)))
6259, 61ovresd 7599 . . . . . . . 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 4214 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (Base‘𝐷))
6461elin2d 4214 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (Base‘𝐷))
658, 7, 38, 63, 64homfval 17736 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6662, 65eqtrd 2774 . . . . . . 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 6723 . . . . . 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 257 . . . . 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 17947 . . . 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 2735 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐶) = (Homf𝐶))
71 eqidd 2735 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐶) = (compf𝐶))
727ressinbas 17290 . . . . . . . . . . 11 (𝑆𝑉 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7348, 72syl 17 . . . . . . . . . 10 (𝜑 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7431, 73eqtrid 2786 . . . . . . . . 9 (𝜑𝐸 = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7574fveq2d 6910 . . . . . . . 8 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
76 eqid 2734 . . . . . . . . . 10 (𝐷s (𝑆 ∩ (Base‘𝐷))) = (𝐷s (𝑆 ∩ (Base‘𝐷)))
77 eqid 2734 . . . . . . . . . 10 (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) = (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))
787, 8, 9, 11, 76, 77fullresc 17901 . . . . . . . . 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 494 . . . . . . . 8 (𝜑 → (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8075, 79eqtrd 2774 . . . . . . 7 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8180adantr 480 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8274fveq2d 6910 . . . . . . . 8 (𝜑 → (compf𝐸) = (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
8378simprd 495 . . . . . . . 8 (𝜑 → (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8482, 83eqtrd 2774 . . . . . . 7 (𝜑 → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8584adantr 480 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
86 df-br 5148 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87 funcrcl 17913 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8886, 87sylbi 217 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8988simpld 494 . . . . . . . . 9 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
90 df-br 5148 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91 funcrcl 17913 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9290, 91sylbi 217 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9392simpld 494 . . . . . . . . 9 (𝐹(𝐶 Func 𝐸)𝐺𝐶 ∈ Cat)
9489, 93jaoi 857 . . . . . . . 8 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
9594elexd 3501 . . . . . . 7 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
9695adantl 481 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
9731ovexi 7464 . . . . . . 7 𝐸 ∈ V
9897a1i 11 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99 ovexd 7465 . . . . . 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 17953 . . . . 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 5158 . . . 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 282 . . 3 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
103102ex 412 . 2 (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)))
1042, 4, 103pm5.21ndd 379 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  ran crn 5689  cres 5690   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  s cress 17273  Hom chom 17308  Catccat 17708  Homf chomf 17710  compfccomf 17711  cat cresc 17855  Subcatcsubc 17856   Func cfunc 17904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-ssc 17857  df-resc 17858  df-subc 17859  df-func 17908
This theorem is referenced by:  fthres2c  17984  fullres2c  17992
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