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

Theorem nat1st2nd 17914
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 17821 . . . 4 Rel (𝐶 Func 𝐷)
3 natrcl.1 . . . . . . 7 𝑁 = (𝐶 Nat 𝐷)
43natrcl 17913 . . . . . 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 494 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 1st2nd 8024 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
82, 6, 7sylancr 586 . . 3 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
95simprd 495 . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 1st2nd 8024 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
112, 9, 10sylancr 586 . . 3 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
128, 11oveq12d 7423 . 2 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
131, 12eleqtrd 2829 1 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cop 4629  Rel wrel 5674  cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973   Func cfunc 17813   Nat cnat 17904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-ixp 8894  df-func 17817  df-nat 17906
This theorem is referenced by:  fuccocl  17929  fuclid  17931  fucrid  17932  fucass  17933  fucsect  17937  invfuc  17939  fucpropd  17942  evlfcllem  18186  evlfcl  18187  curfuncf  18203  yonedalem3a  18239  yonedalem3b  18244  yonedainv  18246  yonffthlem  18247
  Copyright terms: Public domain W3C validator