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Theorem funcres2b 16323
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 4575 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
2 funcrcl 16289 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2sylbi 205 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simpld 473 . . 3 (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat)
54a1i 11 . 2 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐶 ∈ Cat))
6 df-br 4575 . . . . 5 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
7 funcrcl 16289 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
86, 7sylbi 205 . . . 4 (𝐹(𝐶 Func (𝐷cat 𝑅))𝐺 → (𝐶 ∈ Cat ∧ (𝐷cat 𝑅) ∈ Cat))
98simpld 473 . . 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 2606 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
1512, 13, 14subcss1 16268 . . . . . . . 8 (𝜑𝑆 ⊆ (Base‘𝐷))
1611, 15fssd 5953 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘𝐷))
17 eqid 2606 . . . . . . . . . 10 (𝐷cat 𝑅) = (𝐷cat 𝑅)
18 subcrcl 16242 . . . . . . . . . . 11 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1912, 18syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ Cat)
2017, 14, 19, 13, 15rescbas 16255 . . . . . . . . 9 (𝜑𝑆 = (Base‘(𝐷cat 𝑅)))
2120feq3d 5928 . . . . . . . 8 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2211, 21mpbid 220 . . . . . . 7 (𝜑𝐹:𝐴⟶(Base‘(𝐷cat 𝑅)))
2316, 222thd 253 . . . . . 6 (𝜑 → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
2423adantr 479 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐹:𝐴⟶(Base‘𝐷) ↔ 𝐹:𝐴⟶(Base‘(𝐷cat 𝑅))))
25 funcres2b.2 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
2625adantlr 746 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))
27 frn 5949 . . . . . . . . . . . . . . . 16 ((𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))
2826, 27syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))
2912ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 ∈ (Subcat‘𝐷))
3013ad2antrr 757 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 Fn (𝑆 × 𝑆))
31 eqid 2606 . . . . . . . . . . . . . . . 16 (Hom ‘𝐷) = (Hom ‘𝐷)
3211ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝐹:𝐴𝑆)
33 simprl 789 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑥𝐴)
3432, 33ffvelrnd 6250 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑥) ∈ 𝑆)
35 simprr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑦𝐴)
3632, 35ffvelrnd 6250 . . . . . . . . . . . . . . . 16 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (𝐹𝑦) ∈ 𝑆)
3729, 30, 31, 34, 36subcss2 16269 . . . . . . . . . . . . . . 15 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3828, 37sstrd 3574 . . . . . . . . . . . . . 14 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
3938, 282thd 253 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
4039anbi2d 735 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦)))))
41 df-f 5791 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
42 df-f 5791 . . . . . . . . . . . 12 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ ((𝑥𝐺𝑦) Fn (𝑥𝐻𝑦) ∧ ran (𝑥𝐺𝑦) ⊆ ((𝐹𝑥)𝑅(𝐹𝑦))))
4340, 41, 423bitr4g 301 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦))))
4417, 14, 19, 13, 15reschom 16256 . . . . . . . . . . . . . 14 (𝜑𝑅 = (Hom ‘(𝐷cat 𝑅)))
4544ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
4645oveqd 6541 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥)𝑅(𝐹𝑦)) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
4746feq3d 5928 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝑅(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4843, 47bitrd 266 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
4948ralrimivva 2950 . . . . . . . . 9 ((𝜑𝐶 ∈ Cat) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
50 fveq2 6085 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
51 df-ov 6527 . . . . . . . . . . . . . 14 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
5250, 51syl6eqr 2658 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
53 vex 3172 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
54 vex 3172 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5553, 54op1std 7043 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
5655fveq2d 6089 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
5753, 54op2ndd 7044 . . . . . . . . . . . . . . . 16 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
5857fveq2d 6089 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
5956, 58oveq12d 6542 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
60 fveq2 6085 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
61 df-ov 6527 . . . . . . . . . . . . . . 15 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
6260, 61syl6eqr 2658 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
6359, 62oveq12d 6542 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
6452, 63eleq12d 2678 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦))))
65 ovex 6552 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ∈ V
66 ovex 6552 . . . . . . . . . . . . 13 (𝑥𝐻𝑦) ∈ V
6765, 66elmap 7746 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦)))
6864, 67syl6bb 274 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘𝐷)(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘𝐷)(𝐹𝑦))))
6956, 58oveq12d 6542 . . . . . . . . . . . . . 14 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) = ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
7069, 62oveq12d 6542 . . . . . . . . . . . . 13 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
7152, 70eleq12d 2678 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦))))
72 ovex 6552 . . . . . . . . . . . . 13 ((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ∈ V
7372, 66elmap 7746 . . . . . . . . . . . 12 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦)))
7471, 73syl6bb 274 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))(Hom ‘(𝐷cat 𝑅))(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)(Hom ‘(𝐷cat 𝑅))(𝐹𝑦))))
7568, 74bibi12d 333 . . . . . . . . . 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)))))
7675ralxp 5170 . . . . . . . . 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))))
7749, 76sylibr 222 . . . . . . . 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))))
78 ralbi 3046 . . . . . . . 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))))
7977, 78syl 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))))
80793anbi3d 1396 . . . . . 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)))))
81 elixp2 7772 . . . . . 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))))
82 elixp2 7772 . . . . . 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))))
8380, 81, 823bitr4g 301 . . . . 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))))
8412ad2antrr 757 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 ∈ (Subcat‘𝐷))
8513ad2antrr 757 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → 𝑅 Fn (𝑆 × 𝑆))
86 eqid 2606 . . . . . . . . 9 (Id‘𝐷) = (Id‘𝐷)
8711adantr 479 . . . . . . . . . 10 ((𝜑𝐶 ∈ Cat) → 𝐹:𝐴𝑆)
8887ffvelrnda 6249 . . . . . . . . 9 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ 𝑆)
8917, 84, 85, 86, 88subcid 16273 . . . . . . . 8 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → ((Id‘𝐷)‘(𝐹𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥)))
9089eqeq2d 2616 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘(𝐹𝑥)) ↔ ((𝑥𝐺𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘(𝐷cat 𝑅))‘(𝐹𝑥))))
91 eqid 2606 . . . . . . . . . . . . . 14 (comp‘𝐷) = (comp‘𝐷)
9217, 14, 19, 13, 15, 91rescco 16258 . . . . . . . . . . . . 13 (𝜑 → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9392ad2antrr 757 . . . . . . . . . . . 12 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (comp‘𝐷) = (comp‘(𝐷cat 𝑅)))
9493oveqd 6541 . . . . . . . . . . 11 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧)) = (⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧)))
9594oveqd 6541 . . . . . . . . . 10 (((𝜑𝐶 ∈ Cat) ∧ 𝑥𝐴) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘(𝐷cat 𝑅))(𝐹𝑧))((𝑥𝐺𝑦)‘𝑓)))
9695eqeq2d 2616 . . . . . . . . 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))))
97962ralbidv 2968 . . . . . . . 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))))
98972ralbidv 2968 . . . . . . 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))))
9990, 98anbi12d 742 . . . . . 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)))))
10099ralbidva 2964 . . . . 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)))))
10124, 83, 1003anbi123d 1390 . . . 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))))))
102 funcres2b.a . . . . 5 𝐴 = (Base‘𝐶)
103 funcres2b.h . . . . 5 𝐻 = (Hom ‘𝐶)
104 eqid 2606 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
105 eqid 2606 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
106 simpr 475 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐶 ∈ Cat)
10719adantr 479 . . . . 5 ((𝜑𝐶 ∈ Cat) → 𝐷 ∈ Cat)
108102, 14, 103, 31, 104, 86, 105, 91, 106, 107isfunc 16290 . . . 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))))))
109 eqid 2606 . . . . 5 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
110 eqid 2606 . . . . 5 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
111 eqid 2606 . . . . 5 (Id‘(𝐷cat 𝑅)) = (Id‘(𝐷cat 𝑅))
112 eqid 2606 . . . . 5 (comp‘(𝐷cat 𝑅)) = (comp‘(𝐷cat 𝑅))
11317, 12subccat 16274 . . . . . 6 (𝜑 → (𝐷cat 𝑅) ∈ Cat)
114113adantr 479 . . . . 5 ((𝜑𝐶 ∈ Cat) → (𝐷cat 𝑅) ∈ Cat)
115102, 109, 103, 110, 104, 111, 105, 112, 106, 114isfunc 16290 . . . 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))))))
116101, 108, 1153bitr4d 298 . . 3 ((𝜑𝐶 ∈ Cat) → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
117116ex 448 . 2 (𝜑 → (𝐶 ∈ Cat → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺)))
1185, 10, 117pm5.21ndd 367 1 (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺𝐹(𝐶 Func (𝐷cat 𝑅))𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892  Vcvv 3169  wss 3536  cop 4127   class class class wbr 4574   × cxp 5023  ran crn 5026   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  1st c1st 7031  2nd c2nd 7032  𝑚 cmap 7718  Xcixp 7768  Basecbs 15638  Hom chom 15722  compcco 15723  Catccat 16091  Idccid 16092  cat cresc 16234  Subcatcsubc 16235   Func cfunc 16280
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-map 7720  df-pm 7721  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-ress 15645  df-hom 15736  df-cco 15737  df-cat 16095  df-cid 16096  df-homf 16097  df-ssc 16236  df-resc 16237  df-subc 16238  df-func 16284
This theorem is referenced by:  funcres2  16324  funcres2c  16327  fthres2b  16356
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