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Theorem natffn 17456
Description: The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypothesis
Ref Expression
natrcl.1 𝑁 = (𝐶 Nat 𝐷)
Assertion
Ref Expression
natffn 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))

Proof of Theorem natffn
Dummy variables 𝑥 𝑓 𝑦 𝑎 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natrcl.1 . . 3 𝑁 = (𝐶 Nat 𝐷)
2 eqid 2737 . . 3 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2737 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2737 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
5 eqid 2737 . . 3 (comp‘𝐷) = (comp‘𝐷)
61, 2, 3, 4, 5natfval 17453 . 2 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))})
7 ovex 7246 . . . . . . 7 ((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∈ V
87rgenw 3073 . . . . . 6 𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∈ V
9 ixpexg 8603 . . . . . 6 (∀𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∈ V → X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∈ V)
108, 9ax-mp 5 . . . . 5 X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∈ V
1110rabex 5225 . . . 4 {𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ V
1211csbex 5204 . . 3 (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ V
1312csbex 5204 . 2 (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝐶)((𝑟𝑥)(Hom ‘𝐷)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀ ∈ (𝑥(Hom ‘𝐶)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝐷)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝐷)(𝑠𝑦))(𝑎𝑥))} ∈ V
146, 13fnmpoi 7840 1 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408  csb 3811  cop 4547   × cxp 5549   Fn wfn 6375  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  Xcixp 8578  Basecbs 16760  Hom chom 16813  compcco 16814   Func cfunc 17360   Nat cnat 17448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-ixp 8579  df-func 17364  df-nat 17450
This theorem is referenced by:  fuchom  17469  fuchomOLD  17470
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