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| Mirrors > Home > MPE Home > Th. List > mptfvmpt | Structured version Visualization version GIF version | ||
| Description: A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.) |
| Ref | Expression |
|---|---|
| mptfvmpt.y | ⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
| mptfvmpt.g | ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀) |
| mptfvmpt.v | ⊢ 𝑉 = (𝐹‘𝑋) |
| Ref | Expression |
|---|---|
| mptfvmpt | ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mptfvmpt.y | . 2 ⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴)) | |
| 2 | mptfvmpt.g | . 2 ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀) | |
| 3 | mptfvmpt.v | . . . 4 ⊢ 𝑉 = (𝐹‘𝑋) | |
| 4 | 3 | fvexi 6920 | . . 3 ⊢ 𝑉 ∈ V |
| 5 | 4 | mptex 7243 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
| 6 | 1, 2, 5 | fvmpt 7016 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: cidfval 17719 idafval 18102 grpinvfvalALT 18997 grplactfval 19059 odfvalALT 19551 asclfval 21899 ig1pval 26215 ishlg 28610 htthlem 30936 sgnsv 33180 mvrsval 35510 mvhfval 35538 msrfval 35542 lkrfval 39088 pmapfval 39758 watfvalN 39994 ldilfset 40110 ltrnfset 40119 dilfsetN 40154 trnfsetN 40157 trlfset 40162 tgrpfset 40746 tendofset 40760 tendoi 40796 erngfset 40801 erngfset-rN 40809 dvafset 41006 diaffval 41032 dvhfset 41082 docaffvalN 41123 djaffvalN 41135 dibffval 41142 dicffval 41176 dihffval 41232 dihfval 41233 dochffval 41351 djhffval 41398 lcfrlem8 41551 lcdfval 41590 mapdffval 41628 mapdfval 41629 hvmapffval 41760 hdmap1ffval 41797 hdmapffval 41828 hdmapfval 41829 hgmapffval 41887 hgmapfval 41888 hbtlem1 43135 hbtlem7 43137 |
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