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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 2734 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | cofmpt.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) | |
4 | 3 | feqmptd 6958 | . 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
5 | fveq2 6889 | . 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
6 | 1, 2, 4, 5 | fmptco 7124 | 1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5231 ∘ ccom 5680 ⟶wf 6537 ‘cfv 6541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 |
This theorem is referenced by: offsplitfpar 8102 lo1o12 15474 rlimcn1b 15530 mdetralt 22102 tsmsmhm 23642 uniioombllem3 25094 ismbfcn2 25147 itg1climres 25224 iblabslem 25337 iblabs 25338 bddmulibl 25348 limccnp 25400 dvcjbr 25458 dvmptcj 25477 dvef 25489 plypf1 25718 lgamgulmlem2 26524 lgamcvg2 26549 lgseisenlem4 26871 esumcocn 33067 ftc1anclem6 36555 rhmcomulmpl 41122 rhmmpl 41123 selvvvval 41155 evlselv 41157 fundcmpsurbijinjpreimafv 46062 fundcmpsurinjimaid 46066 |
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