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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 6659 | . . 3 ⊢ 𝑉 ∈ V |
5 | 4 | mptex 6963 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
6 | 1, 2, 5 | fvmpt 6745 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: cidfval 16939 idafval 17309 grpinvfvalALT 18135 grplactfval 18192 odfvalALT 18653 asclfval 20565 ig1pval 24773 ishlg 26396 htthlem 28700 sgnsv 30852 mvrsval 32865 mvhfval 32893 msrfval 32897 lkrfval 36383 pmapfval 37052 watfvalN 37288 ldilfset 37404 ltrnfset 37413 dilfsetN 37448 trnfsetN 37451 trlfset 37456 tgrpfset 38040 tendofset 38054 tendoi 38090 erngfset 38095 erngfset-rN 38103 dvafset 38300 diaffval 38326 dvhfset 38376 docaffvalN 38417 djaffvalN 38429 dibffval 38436 dicffval 38470 dihffval 38526 dihfval 38527 dochffval 38645 djhffval 38692 lcfrlem8 38845 lcdfval 38884 mapdffval 38922 mapdfval 38923 hvmapffval 39054 hdmap1ffval 39091 hdmapffval 39122 hdmapfval 39123 hgmapffval 39181 hgmapfval 39182 hbtlem1 40067 hbtlem7 40069 |
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