fuccoval - Metamath Proof Explorer

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Theorem fuccoval 18191

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 18190	. 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 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
10	9	fveq2d 6920	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
11	9	fveq2d 6920	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
12	10, 11	opeq12d 4853	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
13	9	fveq2d 6920	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐻)‘𝑥) = ((1^st ‘𝐻)‘𝑋))
14	12, 13	oveq12d 7469	. . 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 6920	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
16	9	fveq2d 6920	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋))
17	14, 15, 16	oveq123d 7472	. 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 7486	. 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 7032	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⟨cop 4840 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 Basecbs 17431 compcco 17484 Nat cnat 18169 FuncCat cfuc 18170

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-fz 13680 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-hom 17497 df-cco 17498 df-func 18083 df-nat 18171 df-fuc 18172

This theorem is used by: fuccocl 18192 fucass 18196 evlfcllem 18445 yonedalem3b 18503 natcoeq 50677

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