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| Mirrors > Home > MPE Home > Th. List > nat1st2nd | Structured version Visualization version GIF version | ||
| Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| natrcl.1 | ⊢ 𝑁 = (𝐶 Nat 𝐷) | 
| nat1st2nd.2 | ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) | 
| Ref | Expression | 
|---|---|
| nat1st2nd | ⊢ (𝜑 → 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nat1st2nd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) | |
| 2 | relfunc 17908 | . . . 4 ⊢ Rel (𝐶 Func 𝐷) | |
| 3 | natrcl.1 | . . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
| 4 | 3 | natrcl 17999 | . . . . . 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 8065 | . . . 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 8065 | . . . 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 7450 | . 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) | 
| 13 | 1, 12 | eleqtrd 2842 | 1 ⊢ (𝜑 → 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 〈cop 4631 Rel wrel 5689 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 Func cfunc 17900 Nat cnat 17990 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-ixp 8939 df-func 17904 df-nat 17992 | 
| This theorem is referenced by: fuccocl 18013 fuclid 18015 fucrid 18016 fucass 18017 fucsect 18021 invfuc 18023 fucpropd 18026 evlfcllem 18267 evlfcl 18268 curfuncf 18284 yonedalem3a 18320 yonedalem3b 18325 yonedainv 18327 yonffthlem 18328 fuco22nat 49064 fuco22a 49068 fucocolem1 49071 fucocolem3 49073 fucoco 49075 fucolid 49079 fucorid 49080 fucorid2 49081 precofvalALT 49086 precofval2 49087 termcnatval 49193 diag2f1olem 49194 | 
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