Theorem nat1st2nd 17869

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	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))

Assertion

Ref	Expression
nat1st2nd	⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))

Proof of Theorem nat1st2nd

Step	Hyp	Ref	Expression
1		nat1st2nd.2	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
2		relfunc 17777	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
3		natrcl.1	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	3	natrcl 17868	. . . . . 6 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
6	5	simpld 494	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 7980	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
8	2, 6, 7	sylancr 587	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	5	simprd 495	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		1st2nd 7980	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
11	2, 9, 10	sylancr 587	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
12	8, 11	oveq12d 7373	. 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
13	1, 12	eleqtrd 2835	1 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4583  Rel wrel 5626  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929   Func cfunc 17769   Nat cnat 17859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-ixp 8832  df-func 17773  df-nat 17861

This theorem is referenced by:  fuccocl  17882  fuclid  17884  fucrid  17885  fucass  17886  fucsect  17890  invfuc  17892  fucpropd  17895  evlfcllem  18135  evlfcl  18136  curfuncf  18152  yonedalem3a  18188  yonedalem3b  18193  yonedainv  18195  yonffthlem  18196  natoppf2  49391  fuco22nat  49507  fuco22a  49511  fucocolem1  49514  fucocolem3  49516  fucoco  49518  fucolid  49522  fucorid  49523  fucorid2  49524  precofvalALT  49529  precofval2  49530  termcnatval  49696  diag2f1olem  49697  funcsn  49702  0fucterm  49704  concl  49822  coccl  49823  concom  49824  coccom  49825