![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fcod | Structured version Visualization version GIF version |
Description: Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fcod.1 | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
fcod.2 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Ref | Expression |
---|---|
fcod | ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcod.1 | . 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
2 | fcod.2 | . 2 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
3 | fco 6747 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∘ ccom 5682 ⟶wf 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: suppcoss 8213 mapen 9166 mapfienlem3 9431 mapfien 9432 cofsmo 10293 canthp1lem2 10677 gsumval3lem2 19861 psrass1lem 21877 psdmplcl 22086 comet 24435 dvcobr 25890 gsumpart 32782 subfacp1lem5 34794 metakunt33 41689 mapcod 41733 mhmcompl 41781 selvvvval 41818 itcovalendof 47742 |
Copyright terms: Public domain | W3C validator |