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Mirrors > Home > MPE Home > Th. List > fnco | Structured version Visualization version 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 5902 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | fnfun 5902 | . . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
3 | funco 5842 | . . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | |
4 | 1, 2, 3 | syl2an 493 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺)) |
5 | 4 | 3adant3 1074 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → Fun (𝐹 ∘ 𝐺)) |
6 | fndm 5904 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | sseq2d 3596 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ 𝐴)) |
8 | 7 | biimpar 501 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → ran 𝐺 ⊆ dom 𝐹) |
9 | dmcosseq 5308 | . . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
11 | 10 | 3adant2 1073 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = dom 𝐺) |
12 | fndm 5904 | . . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
13 | 12 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom 𝐺 = 𝐵) |
14 | 11, 13 | eqtrd 2644 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → dom (𝐹 ∘ 𝐺) = 𝐵) |
15 | df-fn 5807 | . 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵)) | |
16 | 5, 14, 15 | sylanbrc 695 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊆ wss 3540 dom cdm 5038 ran crn 5039 ∘ ccom 5042 Fun wfun 5798 Fn wfn 5799 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 |
This theorem is referenced by: fco 5971 fnfco 5982 fipreima 8155 cshco 13433 swrdco 13434 isofn 16258 prdsinvlem 17347 prdsmgp 18433 pws1 18439 evlslem1 19336 frlmbas 19918 frlmup3 19958 frlmup4 19959 upxp 21236 uptx 21238 0vfval 26845 xppreima2 28830 psgnfzto1stlem 29181 sseqfv1 29778 sseqfn 29779 sseqfv2 29783 volsupnfl 32624 ftc1anclem5 32659 ftc1anclem8 32662 choicefi 38387 fourierdlem42 39042 |
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