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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 16345 | . . . 4 ⊢ Rel (𝐶 Func 𝐷) | |
3 | natrcl.1 | . . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | 3 | natrcl 16433 | . . . . . 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 474 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
7 | 1st2nd 7105 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
8 | 2, 6, 7 | sylancr 694 | . . 3 ⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
9 | 5 | simprd 478 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
10 | 1st2nd 7105 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) | |
11 | 2, 9, 10 | sylancr 694 | . . 3 ⊢ (𝜑 → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
12 | 8, 11 | oveq12d 6567 | . 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
13 | 1, 12 | eleqtrd 2690 | 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 1475 ∈ wcel 1977 〈cop 4131 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Func cfunc 16337 Nat cnat 16424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-ixp 7795 df-func 16341 df-nat 16426 |
This theorem is referenced by: fuccocl 16447 fuclid 16449 fucrid 16450 fucass 16451 fucsect 16455 invfuc 16457 fucpropd 16460 evlfcllem 16684 evlfcl 16685 curfuncf 16701 yonedalem3a 16737 yonedalem3b 16742 yonedainv 16744 yonffthlem 16745 |
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