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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 6229 | . 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) | |
2 | fnco 6335 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) | |
3 | 2 | 3expb 1113 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵) |
4 | 1, 3 | sylan2b 593 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ⊆ wss 3859 ran crn 5444 ∘ ccom 5447 Fn wfn 6220 ⟶wf 6221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-fun 6227 df-fn 6228 df-f 6229 |
This theorem is referenced by: cocan1 6912 cocan2 6913 ofco 7287 1stcof 7575 2ndcof 7576 axcc3 9706 dmaf 17138 cdaf 17139 gsumzaddlem 18761 prdstopn 21920 xpstopnlem2 22103 prdstgpd 22416 prdsxmslem2 22822 uniiccdif 23862 uniiccvol 23864 uniioombllem2 23867 resinf1o 24801 jensen 25248 occllem 28771 nlelchi 29529 hmopidmchi 29619 iprodefisumlem 32580 brcoffn 39865 brcofffn 39866 stoweidlem27 41854 isomushgr 43473 |
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