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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 6567 | . 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) | |
2 | fnco 6687 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) | |
3 | 2 | 3expb 1119 | . 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 3963 ran crn 5690 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: cocan1 7311 cocan2 7312 coof 7721 ofco 7722 1stcof 8043 2ndcof 8044 axcc3 10476 dmaf 18103 cdaf 18104 gsumzaddlem 19954 prdstopn 23652 xpstopnlem2 23835 prdstgpd 24149 prdsxmslem2 24558 uniiccdif 25627 uniiccvol 25629 uniioombllem2 25632 resinf1o 26593 jensen 27047 occllem 31332 nlelchi 32090 hmopidmchi 32180 1arithidomlem2 33544 iprodefisumlem 35720 brcoffn 44020 brcofffn 44021 stoweidlem27 45983 gricushgr 47824 |
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