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| Mirrors > Home > MPE Home > Th. List > fnfco | Structured version Visualization version GIF version | ||
| Description: Composition of two functions. (Contributed by NM, 22-May-2006.) |
| Ref | Expression |
|---|---|
| fnfco | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-f 6565 | . 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) | |
| 2 | fnco 6686 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) | |
| 3 | 2 | 3expb 1121 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵) |
| 4 | 1, 3 | sylan2b 594 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3951 ran crn 5686 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: cocan1 7311 cocan2 7312 coof 7721 ofco 7722 1stcof 8044 2ndcof 8045 axcc3 10478 dmaf 18094 cdaf 18095 gsumzaddlem 19939 prdstopn 23636 xpstopnlem2 23819 prdstgpd 24133 prdsxmslem2 24542 uniiccdif 25613 uniiccvol 25615 uniioombllem2 25618 resinf1o 26578 jensen 27032 occllem 31322 nlelchi 32080 hmopidmchi 32170 1arithidomlem2 33564 iprodefisumlem 35740 brcoffn 44043 brcofffn 44044 stoweidlem27 46042 gricushgr 47886 fucoid 49043 |
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