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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 16723 | . . . 4 ⊢ Rel (𝐶 Func 𝐷) | |
3 | natrcl.1 | . . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | 3 | natrcl 16811 | . . . . . 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 477 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
7 | 1st2nd 7381 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
8 | 2, 6, 7 | sylancr 698 | . . 3 ⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
9 | 5 | simprd 482 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
10 | 1st2nd 7381 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) | |
11 | 2, 9, 10 | sylancr 698 | . . 3 ⊢ (𝜑 → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
12 | 8, 11 | oveq12d 6831 | . 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
13 | 1, 12 | eleqtrd 2841 | 1 ⊢ (𝜑 → 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 Rel wrel 5271 ‘cfv 6049 (class class class)co 6813 1st c1st 7331 2nd c2nd 7332 Func cfunc 16715 Nat cnat 16802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-ixp 8075 df-func 16719 df-nat 16804 |
This theorem is referenced by: fuccocl 16825 fuclid 16827 fucrid 16828 fucass 16829 fucsect 16833 invfuc 16835 fucpropd 16838 evlfcllem 17062 evlfcl 17063 curfuncf 17079 yonedalem3a 17115 yonedalem3b 17120 yonedainv 17122 yonffthlem 17123 |
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