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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 2737 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | cofmpt.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐶⟶𝐷) | |
4 | 3 | feqmptd 6758 | . 2 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐶 ↦ (𝐹‘𝑦))) |
5 | fveq2 6695 | . 2 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
6 | 1, 2, 4, 5 | fmptco 6922 | 1 ⊢ (𝜑 → (𝐹 ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5120 ∘ ccom 5540 ⟶wf 6354 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: offsplitfpar 7866 lo1o12 15059 rlimcn1b 15115 mdetralt 21459 tsmsmhm 22997 uniioombllem3 24436 ismbfcn2 24489 itg1climres 24566 iblabslem 24679 iblabs 24680 bddmulibl 24690 limccnp 24742 dvcjbr 24800 dvmptcj 24819 dvef 24831 plypf1 25060 lgamgulmlem2 25866 lgamcvg2 25891 lgseisenlem4 26213 esumcocn 31714 ftc1anclem6 35541 fundcmpsurbijinjpreimafv 44475 fundcmpsurinjimaid 44479 |
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