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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 16498 | . . . 4 ⊢ coda = (2nd ∘ 1st ) | |
2 | 1 | fveq1i 6104 | . . 3 ⊢ (coda‘𝐹) = ((2nd ∘ 1st )‘𝐹) |
3 | fo1st 7079 | . . . . 5 ⊢ 1st :V–onto→V | |
4 | fof 6028 | . . . . 5 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 1st :V⟶V |
6 | elex 3185 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V) | |
7 | fvco3 6185 | . . . 4 ⊢ ((1st :V⟶V ∧ 𝐹 ∈ V) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) | |
8 | 5, 6, 7 | sylancr 694 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((2nd ∘ 1st )‘𝐹) = (2nd ‘(1st ‘𝐹))) |
9 | 2, 8 | syl5eq 2656 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = (2nd ‘(1st ‘𝐹))) |
10 | homahom.h | . . . . . 6 ⊢ 𝐻 = (Homa‘𝐶) | |
11 | 10 | homarel 16509 | . . . . 5 ⊢ Rel (𝑋𝐻𝑌) |
12 | 1st2ndbr 7108 | . . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) | |
13 | 11, 12 | mpan 702 | . . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1st ‘𝐹)(𝑋𝐻𝑌)(2nd ‘𝐹)) |
14 | 10 | homa1 16510 | . . . 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 6107 | . 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘(1st ‘𝐹)) = (2nd ‘〈𝑋, 𝑌〉)) |
17 | eqid 2610 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
18 | 10, 17 | homarcl2 16508 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
19 | op2ndg 7072 | . . 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 2648 | 1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ∘ ccom 5042 Rel wrel 5043 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 codaccoda 16494 Homachoma 16496 |
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-1st 7059 df-2nd 7060 df-coda 16498 df-homa 16499 |
This theorem is referenced by: arwhoma 16518 idacd 16535 homdmcoa 16540 coaval 16541 |
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