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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 7007. (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 7007 | . 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 1536 ∈ wcel 2105 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 |
This theorem is referenced by: opco1 8146 opco2 8147 suppcoss 8230 wemapwe 9734 canthp1lem2 10690 yonedainv 18337 frgpcyg 21609 psdmplcl 22183 rhmcomulmpl 22401 comet 24541 dvcobr 25997 gsumpart 33042 pmtrcnel 33091 elrgspnlem1 33231 subfacp1lem5 35168 aks5lem3a 42170 metakunt33 42218 rhmcomulpsr 42537 selvvvval 42571 evlselv 42573 extoimad 44153 imo72b2lem0 44154 imo72b2lem1 44158 fcores 47016 fcoresf1lem 47017 grimco 47817 uspgrlimlem3 47892 |
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