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| Mirrors > Home > MPE Home > Th. List > fvco3d | Structured version Visualization version GIF version | ||
| Description: Value of a function composition. Deduction form of fvco3 7008. (Contributed by Stanislas Polu, 9-Mar-2020.) |
| Ref | Expression |
|---|---|
| fvco3d.1 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
| fvco3d.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| fvco3d | ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvco3d.1 | . 2 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
| 2 | fvco3d.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
| 3 | fvco3 7008 | . 2 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: opco1 8148 opco2 8149 suppcoss 8232 wemapwe 9737 canthp1lem2 10693 yonedainv 18326 frgpcyg 21592 psdmplcl 22166 rhmcomulmpl 22386 comet 24526 dvcobr 25983 gsumpart 33060 pmtrcnel 33109 elrgspnlem1 33246 subfacp1lem5 35189 aks5lem3a 42190 metakunt33 42238 rhmcomulpsr 42561 selvvvval 42595 evlselv 42597 extoimad 44177 imo72b2lem0 44178 imo72b2lem1 44182 fcores 47079 fcoresf1lem 47080 grimco 47880 uspgrlimlem3 47957 fuco111x 49026 fuco112xa 49028 fuco11idx 49030 fuco22natlem3 49039 fuco22natlem 49040 fucoid 49043 fucocolem4 49051 fucolid 49056 fucorid 49057 precofvallem 49061 |
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