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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 6761 | . 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 5693 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: suppcoss 8231 mapen 9180 mapfienlem3 9445 mapfien 9446 cofsmo 10307 canthp1lem2 10691 gsumval3lem2 19939 psrass1lem 21970 psdmplcl 22184 mhmcompl 22400 comet 24542 dvcobr 25998 wrdpmcl 32907 gsumpart 33043 elrgspnlem1 33232 1arithidomlem2 33544 1arithidom 33545 subfacp1lem5 35169 metakunt33 42219 mapcod 42263 mhmcopsr 42536 selvvvval 42572 itcovalendof 48519 |
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