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Mirrors > Home > MPE Home > Th. List > funcofd | Structured version Visualization version GIF version |
Description: Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.) |
Ref | Expression |
---|---|
funcofd.1 | β’ (π β Fun πΉ) |
funcofd.2 | β’ (π β Fun πΊ) |
Ref | Expression |
---|---|
funcofd | β’ (π β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcofd.1 | . . 3 β’ (π β Fun πΉ) | |
2 | fdmrn 6760 | . . 3 β’ (Fun πΉ β πΉ:dom πΉβΆran πΉ) | |
3 | 1, 2 | sylib 217 | . 2 β’ (π β πΉ:dom πΉβΆran πΉ) |
4 | funcofd.2 | . 2 β’ (π β Fun πΊ) | |
5 | fcof 6751 | . 2 β’ ((πΉ:dom πΉβΆran πΉ β§ Fun πΊ) β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran πΉ) | |
6 | 3, 4, 5 | syl2anc 582 | 1 β’ (π β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β‘ccnv 5681 dom cdm 5682 ran crn 5683 β cima 5685 β ccom 5686 Fun wfun 6547 βΆwf 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 |
This theorem is referenced by: smfco 46219 |
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