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Theorem nat1st2nd 18019

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 17926	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
3		natrcl.1	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	3	natrcl 18018	. . . . . 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 8080	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
8	2, 6, 7	sylancr 586	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
9	5	simprd 495	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		1st2nd 8080	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
11	2, 9, 10	sylancr 586	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
12	8, 11	oveq12d 7466	. 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
13	1, 12	eleqtrd 2846	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 395 = wceq 1537 ∈ wcel 2108 ⟨cop 4654 Rel wrel 5705 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 Func cfunc 17918 Nat cnat 18009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-ixp 8956 df-func 17922 df-nat 18011

This theorem is referenced by: fuccocl 18034 fuclid 18036 fucrid 18037 fucass 18038 fucsect 18042 invfuc 18044 fucpropd 18047 evlfcllem 18291 evlfcl 18292 curfuncf 18308 yonedalem3a 18344 yonedalem3b 18349 yonedainv 18351 yonffthlem 18352

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