Theorem natrcl 17292

Description: Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypothesis

Ref	Expression
natrcl.1	⊢ 𝑁 = (𝐶 Nat 𝐷)

Assertion

Ref	Expression
natrcl	⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))

Proof of Theorem natrcl

Dummy variables 𝑥 𝑓 𝑦 𝑎 𝑔 ℎ 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natrcl.1	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		eqid 2758	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2758	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2758	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2758	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
6	1, 2, 3, 4, 5	natfval 17288	. 2 ⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ℎ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝐷)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝐷)(𝑠‘𝑦))(𝑎‘𝑥))})
7	6	elmpocl 7389	1 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  ⦋csb 3807  ⟨cop 4531  ‘cfv 6340  (class class class)co 7156  1^st c1st 7697  2^nd c2nd 7698  Xcixp 8492  Basecbs 16554  Hom chom 16647  compcco 16648   Func cfunc 17196   Nat cnat 17283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-ixp 8493  df-func 17200  df-nat 17285

This theorem is referenced by:  nat1st2nd  17293  natixp  17294  nati  17297  fucco  17304  fuccocl  17306  fuclid  17308  fucrid  17309  fucass  17310