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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 17847 | . . . 4 ⊢ Rel (𝐶 Func 𝐷) | |
3 | natrcl.1 | . . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | 3 | natrcl 17939 | . . . . . 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 493 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
7 | 1st2nd 8041 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) | |
8 | 2, 6, 7 | sylancr 585 | . . 3 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩) |
9 | 5 | simprd 494 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
10 | 1st2nd 8041 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩) | |
11 | 2, 9, 10 | sylancr 585 | . . 3 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩) |
12 | 8, 11 | oveq12d 7434 | . 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)) |
13 | 1, 12 | eleqtrd 2827 | 1 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟨cop 4630 Rel wrel 5677 ‘cfv 6543 (class class class)co 7416 1st c1st 7989 2nd c2nd 7990 Func cfunc 17839 Nat cnat 17930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-ixp 8915 df-func 17843 df-nat 17932 |
This theorem is referenced by: fuccocl 17955 fuclid 17957 fucrid 17958 fucass 17959 fucsect 17963 invfuc 17965 fucpropd 17968 evlfcllem 18212 evlfcl 18213 curfuncf 18229 yonedalem3a 18265 yonedalem3b 18270 yonedainv 18272 yonffthlem 18273 |
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