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Mirrors > Home > MPE Home > Th. List > cofmpt | Structured version Visualization version 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 2822 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | cofmpt.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) | |
4 | 3 | feqmptd 6727 | . 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
5 | fveq2 6664 | . 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
6 | 1, 2, 4, 5 | fmptco 6885 | 1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5138 ∘ 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-csb 3883 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-mpt 5139 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: offsplitfpar 7809 lo1o12 14884 rlimcn1b 14940 mdetralt 21211 tsmsmhm 22748 uniioombllem3 24180 ismbfcn2 24233 itg1climres 24309 iblabslem 24422 iblabs 24423 bddmulibl 24433 limccnp 24483 dvcjbr 24540 dvmptcj 24559 dvef 24571 plypf1 24796 lgamgulmlem2 25601 lgamcvg2 25626 lgseisenlem4 25948 esumcocn 31334 ftc1anclem6 34966 fundcmpsurbijinjpreimafv 43561 fundcmpsurinjimaid 43565 |
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