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Mirrors > Home > ILE Home > Th. List > fnco | GIF version |
Description: Composition of two functions. (Contributed by NM, 22-May-2006.) |
Ref | Expression |
---|---|
fnco | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 4996 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | fnfun 4996 | . . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
3 | funco 4940 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2an 273 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺)) |
5 | 4 | 3adant3 924 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → Fun (𝐹 ∘ 𝐺)) |
6 | fndm 4998 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | sseq2d 2973 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ 𝐴)) |
8 | 7 | biimpar 281 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → ran 𝐺 ⊆ dom 𝐹) |
9 | dmcosseq 4603 | . . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺) | |
10 | 8, 9 | syl 14 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
11 | 10 | 3adant2 923 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
12 | fndm 4998 | . . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
13 | 12 | 3ad2ant2 926 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵) |
14 | 11, 13 | eqtrd 2072 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = 𝐵) |
15 | df-fn 4905 | . 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵)) | |
16 | 5, 14, 15 | sylanbrc 394 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 = wceq 1243 ⊆ wss 2917 dom cdm 4345 ran crn 4346 ∘ ccom 4349 Fun wfun 4896 Fn wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 |
This theorem is referenced by: fco 5056 fnfco 5065 |
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