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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 6688 | . . 3 ⊢ 𝑉 ∈ V |
5 | 4 | mptex 6996 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
6 | 1, 2, 5 | fvmpt 6775 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5110 ‘cfv 6339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 |
This theorem is referenced by: cidfval 17050 idafval 17429 grpinvfvalALT 18261 grplactfval 18318 odfvalALT 18779 asclfval 20692 ig1pval 24925 ishlg 26548 htthlem 28852 sgnsv 31004 mvrsval 33038 mvhfval 33066 msrfval 33070 lkrfval 36724 pmapfval 37393 watfvalN 37629 ldilfset 37745 ltrnfset 37754 dilfsetN 37789 trnfsetN 37792 trlfset 37797 tgrpfset 38381 tendofset 38395 tendoi 38431 erngfset 38436 erngfset-rN 38444 dvafset 38641 diaffval 38667 dvhfset 38717 docaffvalN 38758 djaffvalN 38770 dibffval 38777 dicffval 38811 dihffval 38867 dihfval 38868 dochffval 38986 djhffval 39033 lcfrlem8 39186 lcdfval 39225 mapdffval 39263 mapdfval 39264 hvmapffval 39395 hdmap1ffval 39432 hdmapffval 39463 hdmapfval 39464 hgmapffval 39522 hgmapfval 39523 hbtlem1 40520 hbtlem7 40522 |
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