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Theorem funcres2b 16942
 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 4887 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2 funcrcl 16908 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2sylbi 209 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simpld 490 . . 3 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
54a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat))
6 df-br 4887 . . . . 5 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
7 funcrcl 16908 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
86, 7sylbi 209 . . . 4 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
98simpld 490 . . 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 2778 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
1512, 13, 14subcss1 16887 . . . . . . . 8 (𝜑𝑆 ⊆ (Base‘𝐷))
1611, 15fssd 6305 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘𝐷))
17 eqid 2778 . . . . . . . . . 10 (𝐷cat 𝑅) = (𝐷cat 𝑅)
18 subcrcl 16861 . . . . . . . . . . 11 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1912, 18syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ Cat)
2017, 14, 19, 13, 15rescbas 16874 . . . . . . . . 9 (𝜑𝑆 = (Base‘(𝐷cat 𝑅)))
2120feq3d 6278 . . . . . . . 8 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2211, 21mpbid 224 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘(𝐷cat 𝑅)))
2316, 222thd 257 . . . . . 6 (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2423adantr 474 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
25 funcres2b.2 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
2625adantlr 705 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
2726frnd 6298 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))
2812ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
2913ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
30 eqid 2778 . . . . . . . . . . . . . . . 16 (Hom ‘𝐷) = (Hom ‘𝐷)
3111ad2antrr 716 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴𝑆)
32 simprl 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
3331, 32ffvelrnd 6624 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ 𝑆)
34 simprr 763 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
3531, 34ffvelrnd 6624 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ 𝑆)
3628, 29, 30, 33, 35subcss2 16888 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3727, 36sstrd 3831 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3837, 272thd 257 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
3938anbi2d 622 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))))
40 df-f 6139 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
41 df-f 6139 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
4239, 40, 413bitr4g 306 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦))))
4317, 14, 19, 13, 15reschom 16875 . . . . . . . . . . . . . 14 (𝜑𝑅 = (Hom ‘(𝐷cat 𝑅)))
4443ad2antrr 716 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
4544oveqd 6939 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
4645feq3d 6278 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4742, 46bitrd 271 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4847ralrimivva 3153 . . . . . . . . 9 ((𝜑𝐶 ∈ Cat) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
49 fveq2 6446 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
50 df-ov 6925 . . . . . . . . . . . . . 14 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
5149, 50syl6eqr 2832 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
52 vex 3401 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
53 vex 3401 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5452, 53op1std 7455 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
5554fveq2d 6450 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
5652, 53op2ndd 7456 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
5756fveq2d 6450 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
5855, 57oveq12d 6940 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
59 fveq2 6446 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
60 df-ov 6925 . . . . . . . . . . . . . . 15 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
6159, 60syl6eqr 2832 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
6258, 61oveq12d 6940 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
6351, 62eleq12d 2853 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦))))
64 ovex 6954 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ∈ V
65 ovex 6954 . . . . . . . . . . . . 13 (𝑥𝐻𝑦) ∈ V
6664, 65elmap 8169 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6763, 66syl6bb 279 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
6855, 57oveq12d 6940 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
6968, 61oveq12d 6940 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
7051, 69eleq12d 2853 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦))))
71 ovex 6954 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ∈ V
7271, 65elmap 8169 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
7370, 72syl6bb 279 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
7467, 73bibi12d 337 . . . . . . . . . 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 5509 . . . . . . . . 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 226 . . . . . . . 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 3254 . . . . . . . 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 1515 . . . . . 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 8198 . . . . . 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 8198 . . . . . 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 306 . . . . 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 716 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 ∈ (Subcat‘𝐷))
8413ad2antrr 716 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 Fn (𝑆 × 𝑆))
85 eqid 2778 . . . . . . . . 9 (Id‘𝐷) = (Id‘𝐷)
8611adantr 474 . . . . . . . . . 10 ((𝜑𝐶 ∈ Cat) → 𝐹:𝐴𝑆)
8786ffvelrnda 6623 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑆)
8817, 83, 84, 85, 87subcid 16892 . . . . . . . 8 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → ((Id‘𝐷)‘(𝐹𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)))
8988eqeq2d 2788 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ↔ ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥))))
90 eqid 2778 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
9117, 14, 19, 13, 15, 90rescco 16877 . . . . . . . . . . . . 13 (𝜑 → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9291ad2antrr 716 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9392oveqd 6939 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧)))
9493oveqd 6939 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))
9594eqeq2d 2788 . . . . . . . . 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 3171 . . . . . . . 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 3171 . . . . . . 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 624 . . . . . 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 3167 . . . . 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 1509 . . . 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 2778 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
104 eqid 2778 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
105 simpr 479 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐶 ∈ Cat)
10619adantr 474 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐷 ∈ Cat)
107101, 14, 102, 30, 103, 85, 104, 90, 105, 106isfunc 16909 . . . 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 2778 . . . . 5 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
109 eqid 2778 . . . . 5 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
110 eqid 2778 . . . . 5 (Id‘(𝐷cat 𝑅)) = (Id‘(𝐷cat 𝑅))
111 eqid 2778 . . . . 5 (comp‘(𝐷cat 𝑅)) = (comp‘(𝐷cat 𝑅))
11217, 12subccat 16893 . . . . . 6 (𝜑 → (𝐷cat 𝑅) ∈ Cat)
113112adantr 474 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐷cat 𝑅) ∈ Cat)
114101, 108, 102, 109, 103, 110, 104, 111, 105, 113isfunc 16909 . . . 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 303 . . 3 ((𝜑𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
116115ex 403 . 2 (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺)))
1175, 10, 116pm5.21ndd 371 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  Vcvv 3398   ⊆ wss 3792  ⟨cop 4404   class class class wbr 4886   × cxp 5353  ran crn 5356   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444   ↑𝑚 cmap 8140  Xcixp 8194  Basecbs 16255  Hom chom 16349  compcco 16350  Catccat 16710  Idccid 16711   ↾cat cresc 16853  Subcatcsubc 16854   Func cfunc 16899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-hom 16362  df-cco 16363  df-cat 16714  df-cid 16715  df-homf 16716  df-ssc 16855  df-resc 16856  df-subc 16857  df-func 16903 This theorem is referenced by:  funcres2  16943  funcres2c  16946  fthres2b  16975
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