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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 6701 | . . 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 6692 | . 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 585 | 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 5633 dom cdm 5634 ran crn 5635 β cima 5637 β ccom 5638 Fun wfun 6491 βΆwf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: smfco 45129 |
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