Theorem evlf1 18181

Description: Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlf1.e	⊢ 𝐸 = (𝐶 evalF 𝐷)
evlf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
evlf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
evlf1.b	⊢ 𝐵 = (Base‘𝐶)
evlf1.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
evlf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
evlf1	⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))

Proof of Theorem evlf1

Dummy variables 𝑥 𝑦 𝑓 𝑎 𝑔 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlf1.e	. . . 4 ⊢ 𝐸 = (𝐶 evalF 𝐷)
2		evlf1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		evlf1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		evlf1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
5		eqid 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2729	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
7		eqid 2729	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8	1, 2, 3, 4, 5, 6, 7	evlfval 18178	. . 3 ⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩)
9		ovex 7420	. . . . 5 ⊢ (𝐶 Func 𝐷) ∈ V
10	4	fvexi 6872	. . . . 5 ⊢ 𝐵 ∈ V
11	9, 10	mpoex 8058	. . . 4 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)) ∈ V
12	9, 10	xpex 7729	. . . . 5 ⊢ ((𝐶 Func 𝐷) × 𝐵) ∈ V
13	12, 12	mpoex 8058	. . . 4 ⊢ (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔)))) ∈ V
14	11, 13	op1std 7978	. . 3 ⊢ (𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝐶 Nat 𝐷)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝐷)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩ → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)))
15	8, 14	syl 17	. 2 ⊢ (𝜑 → (1st ‘𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)))
16		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑓 = 𝐹)
17	16	fveq2d 6862	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → (1st ‘𝑓) = (1st ‘𝐹))
18		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → 𝑥 = 𝑋)
19	17, 18	fveq12d 6865	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑥 = 𝑋)) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑋))
20		evlf1.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
21		evlf1.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		fvexd 6873	. 2 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑋) ∈ V)
23	15, 19, 20, 21, 22	ovmpod 7541	1 ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⦋csb 3862  ⟨cop 4595   × cxp 5636  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625   Func cfunc 17816   Nat cnat 17906   eval_F cevlf 18170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-evlf 18174

This theorem is referenced by:  evlfcllem  18182  evlfcl  18183  uncf1  18197  yonedalem3a  18235  yonedalem3b  18240  yonedainv  18242  yonffthlem  18243