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| 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 6759 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∘ ccom 5688 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: suppcoss 8233 mapen 9182 mapfienlem3 9448 mapfien 9449 cofsmo 10310 canthp1lem2 10694 gsumval3lem2 19925 psrass1lem 21953 psdmplcl 22167 mhmcompl 22385 comet 24527 dvcobr 25984 wrdpmcl 32923 gsumpart 33061 elrgspnlem1 33247 1arithidomlem2 33565 1arithidom 33566 subfacp1lem5 35190 metakunt33 42239 mapcod 42284 mhmcopsr 42564 selvvvval 42600 itcovalendof 48595 fucoid 49066 | 
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