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Theorem nat1st2nd 17038

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 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))

**Proof of Theorem nat1st2nd**
Step	Hyp	Ref	Expression
1		nat1st2nd.2	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
2		relfunc 16949	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
3		natrcl.1	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	3	natrcl 17037	. . . . . 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 495	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 7585	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
8	2, 6, 7	sylancr 587	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
9	5	simprd 496	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		1st2nd 7585	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
11	2, 9, 10	sylancr 587	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
12	8, 11	oveq12d 7025	. 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
13	1, 12	eleqtrd 2883	1 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ⟨cop 4472 Rel wrel 5440 ‘cfv 6217 (class class class)co 7007 1^st c1st 7534 2^nd c2nd 7535 Func cfunc 16941 Nat cnat 17028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-oprab 7011 df-mpo 7012 df-1st 7536 df-2nd 7537 df-ixp 8301 df-func 16945 df-nat 17030

This theorem is referenced by: fuccocl 17051 fuclid 17053 fucrid 17054 fucass 17055 fucsect 17059 invfuc 17061 fucpropd 17064 evlfcllem 17288 evlfcl 17289 curfuncf 17305 yonedalem3a 17341 yonedalem3b 17346 yonedainv 17348 yonffthlem 17349

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