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Theorem natixp 18001
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 2736 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
5 natixp.j . . . 4 𝐽 = (Hom ‘𝐷)
6 eqid 2736 . . . 4 (comp‘𝐷) = (comp‘𝐷)
72natrcl 17999 . . . . . . 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 494 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
10 df-br 5143 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylibr 234 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
128simprd 495 . . . . 5 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
13 df-br 5143 . . . . 5 (𝐾(𝐶 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐶 Func 𝐷))
1412, 13sylibr 234 . . . 4 (𝜑𝐾(𝐶 Func 𝐷)𝐿)
152, 3, 4, 5, 6, 11, 14isnat 17996 . . 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 232 . 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 494 1 (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  cop 4631   class class class wbr 5142  cfv 6560  (class class class)co 7432  Xcixp 8938  Basecbs 17248  Hom chom 17309  compcco 17310   Func cfunc 17900   Nat cnat 17990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-ixp 8939  df-func 17904  df-nat 17992
This theorem is referenced by:  natcl  18002  natfn  18003
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