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Mirrors > Home > ILE Home > Th. List > cofmpt | GIF version |
Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
Ref | Expression |
---|---|
cofmpt.1 | ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) |
cofmpt.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
cofmpt | ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofmpt.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | eqidd 2171 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | cofmpt.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) | |
4 | 3 | feqmptd 5547 | . 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
5 | fveq2 5494 | . 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
6 | 1, 2, 4, 5 | fmptco 5660 | 1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ↦ cmpt 4048 ∘ ccom 4613 ⟶wf 5192 ‘cfv 5196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 |
This theorem is referenced by: dvcjbr 13431 dvmptcjx 13445 dvef 13447 |
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