fuccoval - Metamath Proof Explorer

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Theorem fuccoval 18141

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 18140	. 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 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑋))
11	9	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑋))
12	10, 11	opeq12d 4841	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑋)⟩)
13	9	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((1^st ‘𝐻)‘𝑥) = ((1^st ‘𝐻)‘𝑋))
14	12, 13	oveq12d 7438	. . 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 6889	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
16	9	fveq2d 6889	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘𝑥) = (𝑅‘𝑋))
17	14, 15, 16	oveq123d 7441	. 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 7455	. 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 7001	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 2145 Vcvv 3451 ⟨cop 4590 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 Basecbs 17387 compcco 17440 Nat cnat 18119 FuncCat cfuc 18120

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-fz 13640 df-struct 17325 df-slot 17360 df-ndx 17372 df-base 17388 df-hom 17452 df-cco 17453 df-func 18033 df-nat 18121 df-fuc 18122

This theorem is used by: fuccocl 18142 fucass 18146 evlfcllem 18395 yonedalem3b 18453

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