fuccoval - Metamath Proof Explorer

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Theorem fuccoval 18150

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 18149	. 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 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
11	9	fveq2d 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
12	10, 11	opeq12d 4847	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
13	9	fveq2d 6894	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐻)‘𝑥) = ((1^st ‘𝐻)‘𝑋))
14	12, 13	oveq12d 7442	. . 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 6894	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
16	9	fveq2d 6894	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋))
17	14, 15, 16	oveq123d 7445	. 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 7459	. 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 7006	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 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 ‘cfv 6543 (class class class)co 7424 1^st c1st 8003 Basecbs 17396 compcco 17449 Nat cnat 18128 FuncCat cfuc 18129

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-er 8725 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-fz 13647 df-struct 17334 df-slot 17369 df-ndx 17381 df-base 17397 df-hom 17461 df-cco 17462 df-func 18042 df-nat 18130 df-fuc 18131

This theorem is used by: fuccocl 18151 fucass 18155 evlfcllem 18404 yonedalem3b 18462 natcoeq 50482

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