MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcres2c Structured version   Visualization version   GIF version

Theorem funcres2c 17408
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 2737 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2737 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
8 eqid 2737 . . . . . . 7 (Homf𝐷) = (Homf𝐷)
9 funcres2c.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
10 inss2 4144 . . . . . . . 8 (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
1110a1i 11 . . . . . . 7 (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
127, 8, 9, 11fullsubc 17356 . . . . . 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 17196 . . . . . . 7 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15 xpss12 5566 . . . . . . . 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 6500 . . . . . . 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 484 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴𝑆)
2221ffnd 6546 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
2321frnd 6553 . . . . . . 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 17372 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
2625frnd 6553 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
27 eqid 2737 . . . . . . . . . . 11 (Base‘𝐸) = (Base‘𝐸)
28 simpr 488 . . . . . . . . . . 11 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
295, 27, 28funcf1 17372 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
3029frnd 6553 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
31 funcres2c.e . . . . . . . . . 10 𝐸 = (𝐷s 𝑆)
3231, 7ressbasss 16792 . . . . . . . . 9 (Base‘𝐸) ⊆ (Base‘𝐷)
3330, 32sstrdi 3913 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
3426, 33jaodan 958 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
3523, 34ssind 4147 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
36 df-f 6384 . . . . . 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 2737 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
39 simpr 488 . . . . . . . . 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 17374 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
43 eqid 2737 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
44 simpr 488 . . . . . . . . . 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 17374 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
48 funcres2c.r . . . . . . . . . . . . 13 (𝜑𝑆𝑉)
4931, 38resshom 16923 . . . . . . . . . . . . 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 7230 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
5352feq3d 6532 . . . . . . . . 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 958 . . . . . . 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 484 . . . . . . . . . 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, 58ffvelrnd 6905 . . . . . . . . 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, 60ffvelrnd 6905 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (𝑆 ∩ (Base‘𝐷)))
6259, 61ovresd 7375 . . . . . . . 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 4113 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (Base‘𝐷))
6461elin2d 4113 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (Base‘𝐷))
658, 7, 38, 63, 64homfval 17195 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6662, 65eqtrd 2777 . . . . . . 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 6532 . . . . . 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 17403 . . . 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 2738 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐶) = (Homf𝐶))
71 eqidd 2738 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐶) = (compf𝐶))
727ressinbas 16797 . . . . . . . . . . 11 (𝑆𝑉 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7348, 72syl 17 . . . . . . . . . 10 (𝜑 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7431, 73syl5eq 2790 . . . . . . . . 9 (𝜑𝐸 = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7574fveq2d 6721 . . . . . . . 8 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
76 eqid 2737 . . . . . . . . . 10 (𝐷s (𝑆 ∩ (Base‘𝐷))) = (𝐷s (𝑆 ∩ (Base‘𝐷)))
77 eqid 2737 . . . . . . . . . 10 (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) = (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))
787, 8, 9, 11, 76, 77fullresc 17357 . . . . . . . . 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 2777 . . . . . . 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 6721 . . . . . . . 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 2777 . . . . . . 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 5054 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
87 funcrcl 17369 . . . . . . . . . . 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 5054 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
91 funcrcl 17369 . . . . . . . . . . 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 857 . . . . . . . 8 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
9594elexd 3428 . . . . . . 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 7247 . . . . . . 7 𝐸 ∈ V
9897a1i 11 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
99 ovexd 7248 . . . . . 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 17407 . . . . 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 5064 . . . 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 847   = wceq 1543  wcel 2110  Vcvv 3408  cin 3865  wss 3866  cop 4547   class class class wbr 5053   × cxp 5549  ran crn 5552  cres 5553   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  s cress 16784  Hom chom 16813  Catccat 17167  Homf chomf 17169  compfccomf 17170  cat cresc 17313  Subcatcsubc 17314   Func cfunc 17360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-hom 16826  df-cco 16827  df-cat 17171  df-cid 17172  df-homf 17173  df-comf 17174  df-ssc 17315  df-resc 17316  df-subc 17317  df-func 17364
This theorem is referenced by:  fthres2c  17438  fullres2c  17446
  Copyright terms: Public domain W3C validator