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Theorem nat1st2nd 16376

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 16287	. . . 4 ⊢ Rel (𝐶 Func 𝐷)
3		natrcl.1	. . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	3	natrcl 16375	. . . . . 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 473	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 7078	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
8	2, 6, 7	sylancr 693	. . 3 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
9	5	simprd 477	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
10		1st2nd 7078	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
11	2, 9, 10	sylancr 693	. . 3 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
12	8, 11	oveq12d 6541	. 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
13	1, 12	eleqtrd 2685	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 382 = wceq 1474 ∈ wcel 1975 ⟨cop 4126 Rel wrel 5029 ‘cfv 5786 (class class class)co 6523 1^st c1st 7030 2^nd c2nd 7031 Func cfunc 16279 Nat cnat 16366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1711 ax-4 1726 ax-5 1825 ax-6 1873 ax-7 1920 ax-8 1977 ax-9 1984 ax-10 2004 ax-11 2019 ax-12 2031 ax-13 2228 ax-ext 2585 ax-rep 4689 ax-sep 4699 ax-nul 4708 ax-pow 4760 ax-pr 4824 ax-un 6820

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3an 1032 df-tru 1477 df-ex 1695 df-nf 1700 df-sb 1866 df-eu 2457 df-mo 2458 df-clab 2592 df-cleq 2598 df-clel 2601 df-nfc 2735 df-ne 2777 df-ral 2896 df-rex 2897 df-reu 2898 df-rab 2900 df-v 3170 df-sbc 3398 df-csb 3495 df-dif 3538 df-un 3540 df-in 3542 df-ss 3549 df-nul 3870 df-if 4032 df-pw 4105 df-sn 4121 df-pr 4123 df-op 4127 df-uni 4363 df-iun 4447 df-br 4574 df-opab 4634 df-mpt 4635 df-id 4939 df-xp 5030 df-rel 5031 df-cnv 5032 df-co 5033 df-dm 5034 df-rn 5035 df-res 5036 df-ima 5037 df-iota 5750 df-fun 5788 df-fn 5789 df-f 5790 df-f1 5791 df-fo 5792 df-f1o 5793 df-fv 5794 df-ov 6526 df-oprab 6527 df-mpt2 6528 df-1st 7032 df-2nd 7033 df-ixp 7768 df-func 16283 df-nat 16368

This theorem is referenced by: fuccocl 16389 fuclid 16391 fucrid 16392 fucass 16393 fucsect 16397 invfuc 16399 fucpropd 16402 evlfcllem 16626 evlfcl 16627 curfuncf 16643 yonedalem3a 16679 yonedalem3b 16684 yonedainv 16686 yonffthlem 16687

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