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Mirrors > Home > MPE Home > Th. List > homacd | Structured version Visualization version GIF version |
Description: The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
homahom.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
homacd | ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coda 17485 | . . . 4 ⊢ coda = (2nd ∘ 1st ) | |
2 | 1 | fveq1i 6696 | . . 3 ⊢ (coda‘𝐹) = ((2nd ∘ 1st )‘𝐹) |
3 | fo1st 7759 | . . . . 5 ⊢ 1st :V–onto→V | |
4 | fof 6611 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
6 | elex 3416 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
7 | fvco3 6788 | . . . 4 ⊢ ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) | |
8 | 5, 6, 7 | sylancr 590 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) |
9 | 2, 8 | syl5eq 2783 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = (2nd ‘(1st ‘𝐹))) |
10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
11 | 10 | homarel 17496 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
12 | 1st2ndbr 7791 | . . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
13 | 11, 12 | mpan 690 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
14 | 10 | homa1 17497 | . . . 4 ⊢ ((1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹) = 〈𝑋, 𝑌〉) |
16 | 15 | fveq2d 6699 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st ‘𝐹)) = (2nd ‘〈𝑋, 𝑌〉)) |
17 | eqid 2736 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
18 | 10, 17 | homarcl2 17495 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
19 | op2ndg 7752 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
21 | 9, 16, 20 | 3eqtrd 2775 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 class class class wbr 5039 ∘ ccom 5540 Rel wrel 5541 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 Basecbs 16666 codaccoda 17481 Homachoma 17483 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-1st 7739 df-2nd 7740 df-coda 17485 df-homa 17486 |
This theorem is referenced by: arwhoma 17505 idacd 17522 homdmcoa 17527 coaval 17528 |
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