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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 4939 | . . . 4 ⊢ (𝐹 Fn A → Fun 𝐹) | |
2 | fnfun 4939 | . . . 4 ⊢ (𝐺 Fn B → Fun 𝐺) | |
3 | funco 4883 | . . . 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 923 | . 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → Fun (𝐹 ∘ 𝐺)) |
6 | fndm 4941 | . . . . . . 7 ⊢ (𝐹 Fn A → dom 𝐹 = A) | |
7 | 6 | sseq2d 2967 | . . . . . 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 4546 | . . . . 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 922 | . . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
12 | fndm 4941 | . . . 4 ⊢ (𝐺 Fn B → dom 𝐺 = B) | |
13 | 12 | 3ad2ant2 925 | . . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom 𝐺 = B) |
14 | 11, 13 | eqtrd 2069 | . 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom (𝐹 ∘ 𝐺) = B) |
15 | df-fn 4848 | . 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 884 = wceq 1242 ⊆ wss 2911 dom cdm 4288 ran crn 4289 ∘ ccom 4292 Fun wfun 4839 Fn wfn 4840 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 |
This theorem is referenced by: fco 4999 fnfco 5008 |
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