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Theorem natixp 17899
Description: A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1 𝑁 = (𝐶 Nat 𝐷)
natixp.2 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
natixp.b 𝐵 = (Base‘𝐶)
natixp.j 𝐽 = (Hom ‘𝐷)
Assertion
Ref Expression
natixp (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐶   𝑥,𝐾   𝜑,𝑥   𝑥,𝐷   𝑥,𝐿   𝑥,𝐵   𝑥,𝐽
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem natixp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . 3 (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))
2 natrcl.1 . . . 4 𝑁 = (𝐶 Nat 𝐷)
3 natixp.b . . . 4 𝐵 = (Base‘𝐶)
4 eqid 2732 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 natixp.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2732 . . . 4 (comp‘𝐷) = (comp‘𝐷)
72natrcl 17897 . . . . . . 7 (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
81, 7syl 17 . . . . . 6 (𝜑 → (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) ∧ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷)))
98simpld 495 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 5148 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 233 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 496 . . . . 5 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 5148 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
1412, 13sylibr 233 . . . 4 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
152, 3, 4, 5, 6, 11, 14isnat 17894 . . 3 (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥)))))
161, 15mpbid 231 . 2 (𝜑 → (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐷)(𝐾𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(⟨(𝐹𝑥), (𝐾𝑥)⟩(comp‘𝐷)(𝐾𝑦))(𝐴𝑥))))
1716simpld 495 1 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  cop 4633   class class class wbr 5147  cfv 6540  (class class class)co 7405  Xcixp 8887  Basecbs 17140  Hom chom 17204  compcco 17205   Func cfunc 17800   Nat cnat 17888
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-ixp 8888  df-func 17804  df-nat 17890
This theorem is referenced by:  natcl  17900  natfn  17901
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