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Theorem fuccoval 17964

Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fucco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucco.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucco.a	⊢ 𝐴 = (Base‘𝐶)
fucco.o	⊢ · = (comp‘𝐷)
fucco.x	⊢ ∙ = (comp‘𝑄)
fucco.f	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
fucco.g	⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
fuccoval.f	⊢ (𝜑 → 𝑋 ∈ 𝐴)

**Assertion**
Ref	Expression
fuccoval	⊢ (𝜑 → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ · ((1^st ‘𝐻)‘𝑋))(𝑅‘𝑋)))

Proof of Theorem fuccoval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucco.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		fucco.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
3		fucco.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
4		fucco.o	. . 3 ⊢ · = (comp‘𝐷)
5		fucco.x	. . 3 ⊢ ∙ = (comp‘𝑄)
6		fucco.f	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
7		fucco.g	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
8	1, 2, 3, 4, 5, 6, 7	fucco 17963	. 2 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
9		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
10	9	fveq2d 6906	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
11	9	fveq2d 6906	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
12	10, 11	opeq12d 4886	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
13	9	fveq2d 6906	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐻)‘𝑥) = ((1^st ‘𝐻)‘𝑋))
14	12, 13	oveq12d 7444	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥)) = (⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ · ((1^st ‘𝐻)‘𝑋)))
15	9	fveq2d 6906	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
16	9	fveq2d 6906	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋))
17	14, 15, 16	oveq123d 7447	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)) = ((𝑆‘𝑋)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ · ((1^st ‘𝐻)‘𝑋))(𝑅‘𝑋)))
18		fuccoval.f	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19		ovexd 7461	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ · ((1^st ‘𝐻)‘𝑋))(𝑅‘𝑋)) ∈ V)
20	8, 17, 18, 19	fvmptd 7017	1 ⊢ (𝜑 → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩ · ((1^st ‘𝐻)‘𝑋))(𝑅‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⟨cop 4638 ‘cfv 6553 (class class class)co 7426 1^st c1st 7999 Basecbs 17189 compcco 17254 Nat cnat 17940 FuncCat cfuc 17941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-slot 17160 df-ndx 17172 df-base 17190 df-hom 17266 df-cco 17267 df-func 17853 df-nat 17942 df-fuc 17943

This theorem is referenced by: fuccocl 17965 fucass 17969 evlfcllem 18222 yonedalem3b 18280

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