Theorem nat1st2nd 18122

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 18030	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
3		natrcl.1	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	3	natrcl 18121	. . . . . 6 ⊢ (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
5	1, 4	syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
6	5	simpld 500	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 8048	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
8	2, 6, 7	sylancr 599	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	5	simprd 501	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		1st2nd 8048	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
11	2, 9, 10	sylancr 599	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
12	8, 11	oveq12d 7436	. 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
13	1, 12	eleqtrd 2863	1 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  Rel wrel 5656  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998   Func cfunc 18022   Nat cnat 18112

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-ixp 8919  df-func 18026  df-nat 18114

This theorem is used by:  fuccocl  18135  fuclid  18137  fucrid  18138  fucass  18139  fucsect  18143  invfuc  18145  fucpropd  18148  evlfcllem  18388  evlfcl  18389  curfuncf  18405  yonedalem3a  18441  yonedalem3b  18446  yonedainv  18448  yonffthlem  18449  natoppf2  50307  fuco22nat  50423  fuco22a  50427  fucocolem1  50430  fucocolem3  50432  fucoco  50434  fucolid  50438  fucorid  50439  fucorid2  50440  precofvalALT  50445  precofval2  50446  termcnatval  50612  diag2f1olem  50613  funcsn  50618  0fucterm  50620  concl  50738  coccl  50739  concom  50740  coccom  50741