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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 17340 | . . . 4 ⊢ Rel (𝐶 Func 𝐷) | |
3 | natrcl.1 | . . . . . . 7 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
4 | 3 | natrcl 17429 | . . . . . 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 498 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
7 | 1st2nd 7799 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
8 | 2, 6, 7 | sylancr 590 | . . 3 ⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
9 | 5 | simprd 499 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
10 | 1st2nd 7799 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) | |
11 | 2, 9, 10 | sylancr 590 | . . 3 ⊢ (𝜑 → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
12 | 8, 11 | oveq12d 7220 | . 2 ⊢ (𝜑 → (𝐹𝑁𝐺) = (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
13 | 1, 12 | eleqtrd 2836 | 1 ⊢ (𝜑 → 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 〈cop 4537 Rel wrel 5545 ‘cfv 6369 (class class class)co 7202 1st c1st 7748 2nd c2nd 7749 Func cfunc 17332 Nat cnat 17420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-ixp 8568 df-func 17336 df-nat 17422 |
This theorem is referenced by: fuccocl 17445 fuclid 17447 fucrid 17448 fucass 17449 fucsect 17453 invfuc 17455 fucpropd 17458 evlfcllem 17701 evlfcl 17702 curfuncf 17718 yonedalem3a 17754 yonedalem3b 17759 yonedainv 17761 yonffthlem 17762 |
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