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Theorem funcres2b 17163
Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
funcres2b.a 𝐴 = (Base‘𝐶)
funcres2b.h 𝐻 = (Hom ‘𝐶)
funcres2b.r (𝜑𝑅 ∈ (Subcat‘𝐷))
funcres2b.s (𝜑𝑅 Fn (𝑆 × 𝑆))
funcres2b.1 (𝜑𝐹:𝐴𝑆)
funcres2b.2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
Assertion
Ref Expression
funcres2b (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem funcres2b
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5053 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2 funcrcl 17129 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2sylbi 220 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simpld 498 . . 3 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
54a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat))
6 df-br 5053 . . . . 5 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
7 funcrcl 17129 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
86, 7sylbi 220 . . . 4 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
98simpld 498 . . 3 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺𝐶 ∈ Cat)
109a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺𝐶 ∈ Cat))
11 funcres2b.1 . . . . . . . 8 (𝜑𝐹:𝐴𝑆)
12 funcres2b.r . . . . . . . . 9 (𝜑𝑅 ∈ (Subcat‘𝐷))
13 funcres2b.s . . . . . . . . 9 (𝜑𝑅 Fn (𝑆 × 𝑆))
14 eqid 2824 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
1512, 13, 14subcss1 17108 . . . . . . . 8 (𝜑𝑆 ⊆ (Base‘𝐷))
1611, 15fssd 6516 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘𝐷))
17 eqid 2824 . . . . . . . . . 10 (𝐷cat 𝑅) = (𝐷cat 𝑅)
18 subcrcl 17082 . . . . . . . . . . 11 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1912, 18syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ Cat)
2017, 14, 19, 13, 15rescbas 17095 . . . . . . . . 9 (𝜑𝑆 = (Base‘(𝐷cat 𝑅)))
2120feq3d 6489 . . . . . . . 8 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2211, 21mpbid 235 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘(𝐷cat 𝑅)))
2316, 222thd 268 . . . . . 6 (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2423adantr 484 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
25 funcres2b.2 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
2625adantlr 714 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
2726frnd 6509 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))
2812ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
2913ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30 eqid 2824 . . . . . . . . . . . . . . . 16 (Hom ‘𝐷) = (Hom ‘𝐷)
3111ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴𝑆)
32 simprl 770 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
3331, 32ffvelrnd 6840 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ 𝑆)
34 simprr 772 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
3531, 34ffvelrnd 6840 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ 𝑆)
3628, 29, 30, 33, 35subcss2 17109 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3727, 36sstrd 3962 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3837, 272thd 268 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
3938anbi2d 631 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))))
40 df-f 6347 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
41 df-f 6347 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
4239, 40, 413bitr4g 317 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦))))
4317, 14, 19, 13, 15reschom 17096 . . . . . . . . . . . . . 14 (𝜑𝑅 = (Hom ‘(𝐷cat 𝑅)))
4443ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
4544oveqd 7162 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
4645feq3d 6489 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4742, 46bitrd 282 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4847ralrimivva 3186 . . . . . . . . 9 ((𝜑𝐶 ∈ Cat) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
49 fveq2 6658 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50 df-ov 7148 . . . . . . . . . . . . . 14 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
5149, 50syl6eqr 2877 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
52 vex 3483 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
53 vex 3483 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5452, 53op1std 7689 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
5554fveq2d 6662 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
5652, 53op2ndd 7690 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
5756fveq2d 6662 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
5855, 57oveq12d 7163 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
59 fveq2 6658 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60 df-ov 7148 . . . . . . . . . . . . . . 15 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
6159, 60syl6eqr 2877 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
6258, 61oveq12d 7163 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
6351, 62eleq12d 2910 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑m (𝑥𝐻𝑦))))
64 ovex 7178 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ∈ V
65 ovex 7178 . . . . . . . . . . . . 13 (𝑥𝐻𝑦) ∈ V
6664, 65elmap 8425 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6763, 66syl6bb 290 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
6855, 57oveq12d 7163 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
6968, 61oveq12d 7163 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
7051, 69eleq12d 2910 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑m (𝑥𝐻𝑦))))
71 ovex 7178 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ∈ V
7271, 65elmap 8425 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
7370, 72syl6bb 290 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
7467, 73bibi12d 349 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))))
7574ralxp 5699 . . . . . . . . 9 (∀𝑧 ∈ (𝐴 × 𝐴)((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))) ↔ ∀𝑥𝐴𝑦𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
7648, 75sylibr 237 . . . . . . . 8 ((𝜑𝐶 ∈ Cat) → ∀𝑧 ∈ (𝐴 × 𝐴)((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
77 ralbi 3162 . . . . . . . 8 (∀𝑧 ∈ (𝐴 × 𝐴)((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))) → (∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
7876, 77syl 17 . . . . . . 7 ((𝜑𝐶 ∈ Cat) → (∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
79783anbi3d 1439 . . . . . 6 ((𝜑𝐶 ∈ Cat) → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))))
80 elixp2 8455 . . . . . 6 (𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
81 elixp2 8455 . . . . . 6 (𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 × 𝐴) ∧ ∀𝑧 ∈ (𝐴 × 𝐴)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
8279, 80, 813bitr4g 317 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ 𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
8312ad2antrr 725 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 ∈ (Subcat‘𝐷))
8413ad2antrr 725 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85 eqid 2824 . . . . . . . . 9 (Id‘𝐷) = (Id‘𝐷)
8611adantr 484 . . . . . . . . . 10 ((𝜑𝐶 ∈ Cat) → 𝐹:𝐴𝑆)
8786ffvelrnda 6839 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑆)
8817, 83, 84, 85, 87subcid 17113 . . . . . . . 8 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → ((Id‘𝐷)‘(𝐹𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)))
8988eqeq2d 2835 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ↔ ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥))))
90 eqid 2824 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
9117, 14, 19, 13, 15, 90rescco 17098 . . . . . . . . . . . . 13 (𝜑 → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9291ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9392oveqd 7162 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧)))
9493oveqd 7162 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))
9594eqeq2d 2835 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))
96952ralbidv 3194 . . . . . . . 8 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))
97962ralbidv 3194 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) ↔ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))
9889, 97anbi12d 633 . . . . . 6 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → ((((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))) ↔ (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))))
9998ralbidva 3191 . . . . 5 ((𝜑𝐶 ∈ Cat) → (∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))) ↔ ∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))))
10024, 82, 993anbi123d 1433 . . . 4 ((𝜑𝐶 ∈ Cat) → ((𝐹:𝐴⟶(Base‘𝐷) ∧ 𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))) ↔ (𝐹:𝐴⟶(Base‘(𝐷cat 𝑅)) ∧ 𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))))
101 funcres2b.a . . . . 5 𝐴 = (Base‘𝐶)
102 funcres2b.h . . . . 5 𝐻 = (Hom ‘𝐶)
103 eqid 2824 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
104 eqid 2824 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
105 simpr 488 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐶 ∈ Cat)
10619adantr 484 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐷 ∈ Cat)
107101, 14, 102, 30, 103, 85, 104, 90, 105, 106isfunc 17130 . . . 4 ((𝜑𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹:𝐴⟶(Base‘𝐷) ∧ 𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))))
108 eqid 2824 . . . . 5 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
109 eqid 2824 . . . . 5 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
110 eqid 2824 . . . . 5 (Id‘(𝐷cat 𝑅)) = (Id‘(𝐷cat 𝑅))
111 eqid 2824 . . . . 5 (comp‘(𝐷cat 𝑅)) = (comp‘(𝐷cat 𝑅))
11217, 12subccat 17114 . . . . . 6 (𝜑 → (𝐷cat 𝑅) ∈ Cat)
113112adantr 484 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐷cat 𝑅) ∈ Cat)
114101, 108, 102, 109, 103, 110, 104, 111, 105, 113isfunc 17130 . . . 4 ((𝜑𝐶 ∈ Cat) → (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 ↔ (𝐹:𝐴⟶(Base‘(𝐷cat 𝑅)) ∧ 𝐺X𝑧 ∈ (𝐴 × 𝐴)(((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐴 (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)) ∧ ∀𝑦𝐴𝑧𝐴𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓))))))
115100, 107, 1143bitr4d 314 . . 3 ((𝜑𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
116115ex 416 . 2 (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺)))
1175, 10, 116pm5.21ndd 384 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  wss 3919  cop 4555   class class class wbr 5052   × cxp 5540  ran crn 5543   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7145  1st c1st 7677  2nd c2nd 7678  m cmap 8396  Xcixp 8451  Basecbs 16479  Hom chom 16572  compcco 16573  Catccat 16931  Idccid 16932  cat cresc 17074  Subcatcsubc 17075   Func cfunc 17120
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-er 8279  df-map 8398  df-pm 8399  df-ixp 8452  df-en 8500  df-dom 8501  df-sdom 8502  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-9 11700  df-n0 11891  df-z 11975  df-dec 12092  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-hom 16585  df-cco 16586  df-cat 16935  df-cid 16936  df-homf 16937  df-ssc 17076  df-resc 17077  df-subc 17078  df-func 17124
This theorem is referenced by:  funcres2  17164  funcres2c  17167  fthres2b  17196
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