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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 6788 | . . 3 ⊢ 𝑉 ∈ V |
5 | 4 | mptex 7099 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
6 | 1, 2, 5 | fvmpt 6875 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 ‘cfv 6433 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 |
This theorem is referenced by: cidfval 17385 idafval 17772 grpinvfvalALT 18619 grplactfval 18676 odfvalALT 19141 asclfval 21083 ig1pval 25337 ishlg 26963 htthlem 29279 sgnsv 31427 mvrsval 33467 mvhfval 33495 msrfval 33499 lkrfval 37101 pmapfval 37770 watfvalN 38006 ldilfset 38122 ltrnfset 38131 dilfsetN 38166 trnfsetN 38169 trlfset 38174 tgrpfset 38758 tendofset 38772 tendoi 38808 erngfset 38813 erngfset-rN 38821 dvafset 39018 diaffval 39044 dvhfset 39094 docaffvalN 39135 djaffvalN 39147 dibffval 39154 dicffval 39188 dihffval 39244 dihfval 39245 dochffval 39363 djhffval 39410 lcfrlem8 39563 lcdfval 39602 mapdffval 39640 mapdfval 39641 hvmapffval 39772 hdmap1ffval 39809 hdmapffval 39840 hdmapfval 39841 hgmapffval 39899 hgmapfval 39900 hbtlem1 40948 hbtlem7 40950 |
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