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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 6754. (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 6754 | . 2 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 |
This theorem is referenced by: yonedainv 17525 frgpcyg 20714 pmtrcnel 30728 selvval2lem4 39129 extoimad 40508 imo72b2lem0 40509 imo72b2lem1 40514 |
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