MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nat1st2nd Structured version   Visualization version   GIF version

Theorem nat1st2nd 16434
Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1 𝑁 = (𝐶 Nat 𝐷)
nat1st2nd.2 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
Assertion
Ref Expression
nat1st2nd (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))

Proof of Theorem nat1st2nd
StepHypRef Expression
1 nat1st2nd.2 . 2 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
2 relfunc 16345 . . . 4 Rel (𝐶 Func 𝐷)
3 natrcl.1 . . . . . . 7 𝑁 = (𝐶 Nat 𝐷)
43natrcl 16433 . . . . . 6 (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
51, 4syl 17 . . . . 5 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
65simpld 474 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 1st2nd 7105 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
82, 6, 7sylancr 694 . . 3 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
95simprd 478 . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 1st2nd 7105 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
112, 9, 10sylancr 694 . . 3 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
128, 11oveq12d 6567 . 2 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
131, 12eleqtrd 2690 1 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cop 4131  Rel wrel 5043  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   Func cfunc 16337   Nat cnat 16424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-func 16341  df-nat 16426
This theorem is referenced by:  fuccocl  16447  fuclid  16449  fucrid  16450  fucass  16451  fucsect  16455  invfuc  16457  fucpropd  16460  evlfcllem  16684  evlfcl  16685  curfuncf  16701  yonedalem3a  16737  yonedalem3b  16742  yonedainv  16744  yonffthlem  16745
  Copyright terms: Public domain W3C validator