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Theorem funcres2c 16330
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 398 . . 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 397 . . 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 2609 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
7 eqid 2609 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
8 eqid 2609 . . . . . . 7 (Homf𝐷) = (Homf𝐷)
9 funcres2c.d . . . . . . 7 (𝜑𝐷 ∈ Cat)
10 inss2 3795 . . . . . . . 8 (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)
1110a1i 11 . . . . . . 7 (𝜑 → (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷))
127, 8, 9, 11fullsubc 16279 . . . . . 6 (𝜑 → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
1312adantr 479 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))) ∈ (Subcat‘𝐷))
148, 7homffn 16122 . . . . . . 7 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
15 xpss12 5137 . . . . . . . 8 (((𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷) ∧ (𝑆 ∩ (Base‘𝐷)) ⊆ (Base‘𝐷)) → ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷)))
1610, 10, 15mp2an 703 . . . . . . 7 ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))) ⊆ ((Base‘𝐷) × (Base‘𝐷))
17 fnssres 5904 . . . . . . 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 703 . . . . . 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 479 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴𝑆)
22 ffn 5944 . . . . . . 7 (𝐹:𝐴𝑆𝐹 Fn 𝐴)
2321, 22syl 17 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹 Fn 𝐴)
24 frn 5952 . . . . . . . 8 (𝐹:𝐴𝑆 → ran 𝐹𝑆)
2521, 24syl 17 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹𝑆)
26 simpr 475 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
275, 7, 26funcf1 16295 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → 𝐹:𝐴⟶(Base‘𝐷))
28 frn 5952 . . . . . . . . 9 (𝐹:𝐴⟶(Base‘𝐷) → ran 𝐹 ⊆ (Base‘𝐷))
2927, 28syl 17 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐷)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
30 eqid 2609 . . . . . . . . . . 11 (Base‘𝐸) = (Base‘𝐸)
31 simpr 475 . . . . . . . . . . 11 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
325, 30, 31funcf1 16295 . . . . . . . . . 10 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → 𝐹:𝐴⟶(Base‘𝐸))
33 frn 5952 . . . . . . . . . 10 (𝐹:𝐴⟶(Base‘𝐸) → ran 𝐹 ⊆ (Base‘𝐸))
3432, 33syl 17 . . . . . . . . 9 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐸))
35 funcres2c.e . . . . . . . . . 10 𝐸 = (𝐷s 𝑆)
3635, 7ressbasss 15705 . . . . . . . . 9 (Base‘𝐸) ⊆ (Base‘𝐷)
3734, 36syl6ss 3579 . . . . . . . 8 ((𝜑𝐹(𝐶 Func 𝐸)𝐺) → ran 𝐹 ⊆ (Base‘𝐷))
3829, 37jaodan 821 . . . . . . 7 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (Base‘𝐷))
3925, 38ssind 3798 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷)))
40 df-f 5794 . . . . . 6 (𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)) ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ (𝑆 ∩ (Base‘𝐷))))
4123, 39, 40sylanbrc 694 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
42 eqid 2609 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
43 simpr 475 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝐹(𝐶 Func 𝐷)𝐺)
44 simplrl 795 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑥𝐴)
45 simplrr 796 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → 𝑦𝐴)
465, 6, 42, 43, 44, 45funcf2 16297 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
47 eqid 2609 . . . . . . . . . 10 (Hom ‘𝐸) = (Hom ‘𝐸)
48 simpr 475 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝐹(𝐶 Func 𝐸)𝐺)
49 simplrl 795 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑥𝐴)
50 simplrr 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → 𝑦𝐴)
515, 6, 47, 48, 49, 50funcf2 16297 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
52 funcres2c.r . . . . . . . . . . . . 13 (𝜑𝑆𝑉)
5335, 42resshom 15847 . . . . . . . . . . . . 13 (𝑆𝑉 → (Hom ‘𝐷) = (Hom ‘𝐸))
5452, 53syl 17 . . . . . . . . . . . 12 (𝜑 → (Hom ‘𝐷) = (Hom ‘𝐸))
5554ad2antrr 757 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (Hom ‘𝐷) = (Hom ‘𝐸))
5655oveqd 6544 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐸)(𝐹𝑦)))
5756feq3d 5931 . . . . . . . . 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))))
5851, 57mpbird 245 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝐹(𝐶 Func 𝐸)𝐺) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
5946, 58jaodan 821 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6059an32s 841 . . . . . 6 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6141adantr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴⟶(𝑆 ∩ (Base‘𝐷)))
62 simprl 789 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
6361, 62ffvelrnd 6253 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (𝑆 ∩ (Base‘𝐷)))
64 simprr 791 . . . . . . . . . 10 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
6561, 64ffvelrnd 6253 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (𝑆 ∩ (Base‘𝐷)))
6663, 65ovresd 6677 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))(𝐹𝑦)) = ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)))
6710, 63sseldi 3565 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ (Base‘𝐷))
6810, 65sseldi 3565 . . . . . . . . 9 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ (Base‘𝐷))
698, 7, 42, 67, 68homfval 16121 . . . . . . . 8 (((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)(Homf𝐷)(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
7066, 69eqtrd 2643 . . . . . . 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)))
7170feq3d 5931 . . . . . 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))))
7260, 71mpbird 245 . . . . 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)))
735, 6, 13, 19, 41, 72funcres2b 16326 . . . 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))
74 eqidd 2610 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐶) = (Homf𝐶))
75 eqidd 2610 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐶) = (compf𝐶))
767ressinbas 15709 . . . . . . . . . . 11 (𝑆𝑉 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7752, 76syl 17 . . . . . . . . . 10 (𝜑 → (𝐷s 𝑆) = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7835, 77syl5eq 2655 . . . . . . . . 9 (𝜑𝐸 = (𝐷s (𝑆 ∩ (Base‘𝐷))))
7978fveq2d 6092 . . . . . . . 8 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
80 eqid 2609 . . . . . . . . . 10 (𝐷s (𝑆 ∩ (Base‘𝐷))) = (𝐷s (𝑆 ∩ (Base‘𝐷)))
81 eqid 2609 . . . . . . . . . 10 (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) = (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))
827, 8, 9, 11, 80, 81fullresc 16280 . . . . . . . . 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))))))))
8382simpld 473 . . . . . . . 8 (𝜑 → (Homf ‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8479, 83eqtrd 2643 . . . . . . 7 (𝜑 → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8584adantr 479 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (Homf𝐸) = (Homf ‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8678fveq2d 6092 . . . . . . . 8 (𝜑 → (compf𝐸) = (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))))
8782simprd 477 . . . . . . . 8 (𝜑 → (compf‘(𝐷s (𝑆 ∩ (Base‘𝐷)))) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8886, 87eqtrd 2643 . . . . . . 7 (𝜑 → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
8988adantr 479 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (compf𝐸) = (compf‘(𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
90 df-br 4578 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
91 funcrcl 16292 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9290, 91sylbi 205 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9392simpld 473 . . . . . . . . 9 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
94 df-br 4578 . . . . . . . . . . 11 (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
95 funcrcl 16292 . . . . . . . . . . 11 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9694, 95sylbi 205 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐸)𝐺 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
9796simpld 473 . . . . . . . . 9 (𝐹(𝐶 Func 𝐸)𝐺𝐶 ∈ Cat)
9893, 97jaoi 392 . . . . . . . 8 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ Cat)
99 elex 3184 . . . . . . . 8 (𝐶 ∈ Cat → 𝐶 ∈ V)
10098, 99syl 17 . . . . . . 7 ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → 𝐶 ∈ V)
101100adantl 480 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐶 ∈ V)
102 ovex 6555 . . . . . . . 8 (𝐷s 𝑆) ∈ V
10335, 102eqeltri 2683 . . . . . . 7 𝐸 ∈ V
104103a1i 11 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → 𝐸 ∈ V)
105 ovex 6555 . . . . . . 7 (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) ∈ V
106105a1i 11 . . . . . 6 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷))))) ∈ V)
10774, 75, 85, 89, 101, 101, 104, 106funcpropd 16329 . . . . 5 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐶 Func 𝐸) = (𝐶 Func (𝐷cat ((Homf𝐷) ↾ ((𝑆 ∩ (Base‘𝐷)) × (𝑆 ∩ (Base‘𝐷)))))))
108107breqd 4588 . . . 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))
10973, 108bitr4d 269 . . 3 ((𝜑 ∧ (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
110109ex 448 . 2 (𝜑 → ((𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺)))
1112, 4, 110pm5.21ndd 367 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func 𝐸)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  wss 3539  cop 4130   class class class wbr 4577   × cxp 5026  ran crn 5029  cres 5030   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15641  s cress 15642  Hom chom 15725  Catccat 16094  Homf chomf 16096  compfccomf 16097  cat cresc 16237  Subcatcsubc 16238   Func cfunc 16283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-hom 15739  df-cco 15740  df-cat 16098  df-cid 16099  df-homf 16100  df-comf 16101  df-ssc 16239  df-resc 16240  df-subc 16241  df-func 16287
This theorem is referenced by:  fthres2c  16360  fullres2c  16368
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